This deck covers everything linear. It shows a constant rate of change in a table, in a graph, and in an equation, finds slope from two points and reads it as a rate with units, and works through slope-intercept, point-slope, and standard form, along with horizontal and vertical lines and the slopes of parallel and perpendicular lines. It then builds a model from a description or from two data points and predicts with it, and closes with scatter plots, best-fit lines, and correlation. It targets four classic errors: subtracting the coordinates in mismatched order, confusing zero slope with undefined slope, taking only half of a negative reciprocal, and reading the y-intercept off an equation that has not yet been solved for y.
Subject: College Algebra · 142 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
College Algebra - Deck 11
Constant rate of change, slope with units, every form of a line, parallel and perpendicular, and turning a word problem into a model that predicts.
Objectives
A linear function is the one function that changes at the same rate forever. That single fact generates every rule in this deck.
Bring a pencil. Every worked example ends by checking the answer against the original problem, and you should check along with me.
Warm-up
Discussion prompt
Before we open Linear Functions and Modeling: without looking back, what was the main idea of Graphs, Transformations, and Symmetry, and what could you do by the end of it that you could not do before?
Hint: One sentence for the idea, one for the skill. If the second one is blank, that is the part to revisit.
Answer:
That deck is the visual half of College Algebra. It covers plotting and intercepts, the distance and midpoint formulas, and the library of parent-function shapes, then every rigid and non-rigid transformation and the order in which they must be applied, the three symmetry tests, and circles from standard form through completing the square. It targets four classic errors: shifting the wrong way for a horizontal translation, stretching before shifting, mixing up the two reflections, and reading the radius straight off the squared number.
Section
Part 1
Concept
A function is linear when every equal step in the input produces the same change in the output. Not a similar change - the same one, every single time.
linear function — A function whose output changes by a constant amount for each one-unit increase in the input. Its graph is a straight line.
\[ f(x) = mx + b \]
Here m is that constant rate and b is the output when the input is zero.
Counterexample
Discussion prompt
A function is linear when every equal step in the input produces the same change in the output. Not a similar change - the same one, every single time.
That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.
Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.
Intuition
Picture climbing a staircase where every step is exactly the same height. After ten steps you know your height without measuring: ten times the step height, plus wherever you started.
That is the whole idea. The step height is the slope. Where you started is the intercept. A curve is a staircase whose steps keep changing height - that is why curves need calculus and lines do not.
So when a problem says per - dollars per mile, degrees per hour, people per year - your first thought should be: this is linear.
Analogy
Discussion prompt
Explain A staircase with identical steps by analogy to something with no College Algebra in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.
Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.
Answer:
Picture climbing a staircase where every step is exactly the same height. After ten steps you know your height without measuring: ten times the step height, plus wherever you started.
Ranking
Put in order
Put the moves of Worked example: is this table linear? into the order they have to happen.
Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. From 2 to 5 to 8 to 11 the input rises by 3 each time.
Worked example
Decide whether this table could come from a linear function. If it can, write the function.
| x | y |
|---|---|
| 2 | 7 |
| 5 | 1 |
| 8 | -5 |
| 11 | -11 |
Check that the inputs step by the same amount
Why: From 2 to 5 to 8 to 11 the input rises by 3 each time. Equal input steps are what make the output changes comparable.
Find each change in output
Why: From 7 to 1 is a drop of 6. From 1 to negative 5 is a drop of 6. From negative 5 to negative 11 is a drop of 6. The same change every time.
Divide the output change by the input change to get the rate
Why: A drop of 6 for every rise of 3 in the input is a rate of negative 2 per unit.
\[ m = \frac{-6}{3} = -2 \]
Find the starting value
Why: Use one point in the slope-intercept form and solve for the constant. The point where the input is 2 and the output is 7 works.
\[ 7 = -2(2) + b \;\Rightarrow\; 7 = -4 + b \;\Rightarrow\; b = 11 \]
\[ f(x) = -2x + 11 \]
Verify with a point we did not use
Why: Substituting the input 11 gives negative 22 plus 11, which is negative 11 - exactly the last row of the table. The model reproduces the data, so it is right.
\[ f(11) = -2(11) + 11 = -22 + 11 = -11 \quad \checkmark \]
Picture it
Animation
Shows: Each line of the worked example "is this table linear?", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: From 2 to 5 to 8 to 11 the input rises by 3 each time. Equal input steps are what make the output changes comparable.
Concept
Every line is pinned down by two numbers and nothing more: where it starts and how fast it changes.
| Number | Name | What it answers |
|---|---|---|
| m | slope | How much does the output change per one unit of input? |
| b | y-intercept | What is the output when the input is zero? |
Give me those two numbers and I can draw your line, write its equation, and predict any value. Almost every problem in this deck is secretly the task: find m, then find b.
Comparison
Comparison matrix
From A line hands you exactly two facts: refill the What it answers column from what you know. The rest of the table is as it appeared.
| Number | Name | What it answers |
|---|---|---|
| m | slope | How much does the output change per one unit of input? |
| b | y-intercept | What is the output when the input is zero? |
Anomaly
Predict first
A student writes this, and it looks reasonable:
The output changes are 3, then 6, then 3. Not constant - so this table is not linear.
It is wrong. Say what breaks — and say it before you turn the page.
Correct: The output changes were compared without noticing that the input steps were 1, then 2, then 1.
Divide each output change by its own input change. Compare rates, not raw jumps.
Why: The output changes were compared without noticing that the input steps were 1, then 2, then 1. Bigger input steps should produce bigger output changes.
Trap
The output changes are 3, then 6, then 3. Not constant - so this table is not linear.
| x | y |
|---|---|
| 0 | 5 |
| 1 | 8 |
| 3 | 14 |
| 4 | 17 |
The comparison was unfair
Why: The output changes were compared without noticing that the input steps were 1, then 2, then 1. Bigger input steps should produce bigger output changes.
Divide each output change by its own input change. Compare rates, not raw jumps.
| input step | output change | rate |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 6 | 3 |
| 1 | 3 | 3 |
The rate is 3 every time, so the table IS linear
Why: Constant rate is the definition. The starting output is 5, so the function is the one below.
\[ f(x) = 3x + 5 \]
Check the row that was skipped
Why: Substituting the input 3 gives 9 plus 5, which is 14 - the third row exactly.
Pattern
Step through it
Step through Trap: judging a table before checking the input steps one row at a time. What is driving the change, and what would the row after the last one be?
Section
Part 2
Picture it
Figure (svg): A rising line on a pair of axes with a dashed right triangle beneath it: the horizontal leg is labeled run equals 4 and the vertical leg is labeled rise equals 3.
Discussion prompt
Read the picture before the words. What is this showing, and what is the one thing it is built to make obvious? Commit to an answer, then read on.
Hint: Name the parts, then say what changes between them — and if nothing changes, say what is being held still.
Answer:
Walk from one point on a line to another. Rise is how far you went up, run is how far you went across. Slope is the first divided by the second.
Concept
Walk from one point on a line to another. Rise is how far you went up, run is how far you went across. Slope is the first divided by the second.
Figure (svg): A rising line on a pair of axes with a dashed right triangle beneath it: the horizontal leg is labeled run equals 4 and the vertical leg is labeled rise equals 3.
\[ m = \frac{\text{rise}}{\text{run}} = \frac{3}{4} \]
Rise is a signed quantity: going down is a negative rise. Run is measured left to right, so it is positive as long as you read the picture in the usual direction.
Explain it
Discussion prompt
Explain Slope is rise over run to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.
Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.
Answer:
Walk from one point on a line to another. Rise is how far you went up, run is how far you went across. Slope is the first divided by the second.
Intuition
A wheelchair ramp built to code rises about 1 inch for every 12 inches of run. A staircase rises about 7 inches for every 11 inches of run. Both are lines; the staircase is the steeper one because its ratio is larger.
| Surface | rise | run | slope |
|---|---|---|---|
| ramp | 1 in | 12 in | about 0.08 |
| staircase | 7 in | 11 in | about 0.64 |
| ladder | 8 ft | 2 ft | 4 |
Bigger absolute value means steeper. A slope near zero is nearly flat. A huge slope is nearly a wall. You already have this instinct in your legs - slope just puts a number on it.
Pattern
Step through it
Step through Slope is steepness you can feel one row at a time. What is driving the change, and what would the row after the last one be?
Concept
You rarely get to count squares. Usually you get two points, and you subtract to find the rise and the run.
\[ m = \frac{y_2 - y_1}{x_2 - x_1}, \qquad x_1 \ne x_2 \]
slope — The change in output divided by the change in input between any two points on the line. Every pair of points on the same line gives the same value.
The one rule that matters: whichever point you call first in the top must also be the first in the bottom. Consistency, not order, is what the formula demands.
Definition probe
Sort into buckets
Every line below is part of the definition of linear function or of slope — one or the other, never both. Put each where it belongs.
Picture it
Animation
Shows: Slope is a rate with units — a rendered Manim animation.
Rendered with Manim.
Takeaway: Naming the units turns a number into a sentence.
Step zero
Discussion prompt
Worked example: slope through two points — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.
Hint: It starts with: Label the points before touching the formula
Answer:
Worked example
Find the slope of the line through these two points.
\[ (-3,\, 5) \quad \text{and} \quad (2,\, -1) \]
Label the points before touching the formula
Why: Naming them stops the mixed-order mistake. Call the first point one and the second point two, and never swap partway.
\[ (x_1, y_1) = (-3, 5), \qquad (x_2, y_2) = (2, -1) \]
Substitute into the slope formula, keeping every sign
Why: The subtraction of a negative three in the denominator becomes an addition - that is the step where signs usually get lost.
\[ m = \frac{-1 - 5}{2 - (-3)} = \frac{-6}{2 + 3} = \frac{-6}{5} \]
State the slope
Why: A negative slope means the line falls left to right: six down for every five across.
\[ m = -\frac{6}{5} \]
Verify by reversing which point is first
Why: If the answer is really the slope, swapping both subtractions must give the same number. It does - the two negatives cancel.
\[ \frac{5 - (-1)}{-3 - 2} = \frac{6}{-5} = -\frac{6}{5} \quad \checkmark \]
Trap
Find the slope through the points below. The top starts with the second point, the bottom starts with the first.
\[ (1,\, 2) \quad \text{and} \quad (5,\, 10) \]
\[ m = \frac{10 - 2}{1 - 5} = \frac{8}{-4} = -2 \]
The sign came out backwards
Why: The numerator was read second point minus first, but the denominator was read first minus second. Flipping one of the two flips the whole answer.
Fix the order once and hold it for both the top and the bottom.
\[ (x_1, y_1) = (1, 2), \qquad (x_2, y_2) = (5, 10) \]
\[ m = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2 \]
Sanity check against the picture
Why: The output climbed from 2 to 10 while the input climbed from 1 to 5. Both went up, so the slope must be positive. A negative answer was impossible.
Concept
Before you compute anything, look at the line. Its slope has to be one of exactly four kinds, and that alone catches most sign errors.
Figure (svg): Four small panels showing a line rising left to right, a line falling left to right, a flat horizontal line, and a vertical line, labeled positive, negative, zero, and undefined.
| Look of the line | Slope | Equation form |
|---|---|---|
| rises left to right | positive | y = mx + b with m positive |
| falls left to right | negative | y = mx + b with m negative |
| perfectly flat | zero | y = k |
| perfectly vertical | undefined | x = k |
Trade off
Comparison matrix
From Four kinds of slope: every row here is a choice with a cost. Fill the Slope column, then say which row you would actually pick and what you give up for it.
| Look of the line | Slope | Equation form |
|---|---|---|
| rises left to right | positive | y = mx + b with m positive |
| falls left to right | negative | y = mx + b with m negative |
| perfectly flat | zero | y = k |
| perfectly vertical | undefined | x = k |
Estimation
Predict first
Find the slope and the equation of the line through each pair.
Commit before you compute: what does Worked example: the flat line and the vertical line come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.
Correct: Verify both points satisfy each equation
Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Both points in part (a) have output 4, so both satisfy the first equation.
Worked example
Find the slope and the equation of the line through each pair.
\[ \text{(a) } (-1, 4) \text{ and } (6, 4) \qquad \text{(b) } (2, -3) \text{ and } (2, 9) \]
Part (a): subtract the outputs first
Why: The outputs are identical, so the rise is zero. Zero divided by a nonzero run is zero - a real number, and a perfectly legal slope.
\[ m = \frac{4 - 4}{6 - (-1)} = \frac{0}{7} = 0 \]
Write the equation of a horizontal line
Why: The output never changes, so the equation only has to record that constant output. The input is free to be anything.
\[ y = 4 \]
Part (b): now the inputs are identical
Why: The run is zero, and division by zero is not defined. There is no number that measures this steepness, so we say the slope is undefined - not zero.
\[ m = \frac{9 - (-3)}{2 - 2} = \frac{12}{0} \quad \text{undefined} \]
Write the equation of a vertical line
Why: Every point on it has the same input, so the equation constrains the input only. It is a line, but not a function - one input has many outputs.
\[ x = 2 \]
Verify both points satisfy each equation
Why: Both points in part (a) have output 4, so both satisfy the first equation. Both points in part (b) have input 2, so both satisfy the second. Nothing was left out.
Picture it
Animation
Shows: Each line of the worked example "the flat line and the vertical line", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: Both points in part (a) have output 4, so both satisfy the first equation. Both points in part (b) have input 2, so both satisfy the second. Nothing was left out.
Anomaly
Predict first
A student writes this, and it looks reasonable:
A vertical line does not go left or right at all, so nothing changes - its slope must be zero.
It is wrong. Say what breaks — and say it before you turn the page.
Correct: Zero is what you get when the top is zero.
Ask which coordinate stayed the same. Repeated outputs give a flat line; repeated inputs give a vertical line.
Why: Zero is what you get when the top is zero. Zero on the bottom is not a number at all - it is undefined. The two situations are opposites, not the same.
Trap
A vertical line does not go left or right at all, so nothing changes - its slope must be zero.
\[ x = 4 \;\text{through}\; (4, 1) \text{ and } (4, 6) \]
\[ m = \frac{6 - 1}{4 - 4} = \frac{5}{0} \]
Five divided by zero is not zero
Why: Zero is what you get when the top is zero. Zero on the bottom is not a number at all - it is undefined. The two situations are opposites, not the same.
Ask which coordinate stayed the same. Repeated outputs give a flat line; repeated inputs give a vertical line.
\[ y = 4 \;\text{through}\; (1, 4) \text{ and } (6, 4) \]
\[ m = \frac{4 - 4}{6 - 1} = \frac{0}{5} = 0 \]
Zero on top gives slope zero; zero on the bottom gives undefined
Why: Horizontal line: slope zero, equation sets y equal to a constant. Vertical line: slope undefined, equation sets x equal to a constant. Read which letter is trapped.
Notation
Annotate
From Trap: zero slope versus undefined slope — read this one piece at a time. What is each part doing?
On: \( x = 4 \;\text{through}\; (4, 1) \text{ and } (4, 6) \)
Ranking
Put in order
These are the steps of Pattern: computing a slope, every time, scrambled. Put them back in order before the next slide shows you.
Why: This is the order the recipe itself gives. Recalling the sequence without the slide in front of you is the difference between recognising the method and being able to run it — most of what goes wrong in practice is a step done out of turn.
Pattern
In an applied problem, add one more step: attach units. Output units on top, input units on the bottom, read as output units per input unit.
Elimination
Eliminate the wrong options
What is the slope of the candle's height as a function of time, with units?
3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.
Survives elimination: A
Why: The points are (2, 18) and (6, 9). The rise is 9 minus 18, which is negative 9; the run is 6 minus 2, which is 4. So the slope is negative 9 over 4, or negative 2.25 centimeters per hour - the candle loses 2.25 centimeters of height each hour.
Check
A candle burns steadily. Two hours after lighting it measures 18 centimeters tall; six hours after lighting it measures 9 centimeters tall. Work out the rate before you look at the choices.
Check your understanding
What is the slope of the candle's height as a function of time, with units?
Answer: A
Why: The points are (2, 18) and (6, 9). The rise is 9 minus 18, which is negative 9; the run is 6 minus 2, which is 4. So the slope is negative 9 over 4, or negative 2.25 centimeters per hour - the candle loses 2.25 centimeters of height each hour.
Section
Part 3
Concept
The most useful way to write a line is the one that shows you both of its facts at a glance.
\[ y = mx + b \]
slope-intercept form — A line written with the output alone on the left. The coefficient of the input is the slope; the constant added on is the output when the input is zero.
Notice what this form demands: the output must be alone on one side with a coefficient of one. Until that is true, the numbers you see are not the slope and the intercept.
Picture it
Animation
Shows: Two forms, two situations — a rendered Manim animation.
Rendered with Manim.
Takeaway: Point-slope needs less information, so it is usually faster.
Intuition
Read a line in slope-intercept form the way you would read directions to a friend's house: start here, then repeat this move.
| Part of the equation | Direction it gives |
|---|---|
| the constant term | Start on the vertical axis at this height. |
| the coefficient of the input | From there, go up by the top number and right by the bottom number. |
| repeat | Do that move again and again, forwards and backwards. |
You never need a table of values to graph a line. Two points is enough, and this form hands you the first one free.
Picture it
Figure (svg): A coordinate grid with a line rising from the point (0, -4) through (3, -2) and crossing the horizontal axis at (6, 0), with the three points marked.
Discussion prompt
Read the picture before the words. What is this showing, and what is the one thing it is built to make obvious? Commit to an answer, then read on.
Hint: Name the parts, then say what changes between them — and if nothing changes, say what is being held still.
Answer:
Graph this line using only the slope and the intercept.
Worked example
Graph this line using only the slope and the intercept.
\[ y = \frac{2}{3}x - 4 \]
Plot the starting point on the vertical axis
Why: The constant term is negative four, so when the input is zero the output is negative four. That gives the point where the graph crosses the vertical axis.
\[ (0,\, -4) \]
Read the slope as a rise over a run
Why: The coefficient is already a fraction, so the top is the rise and the bottom is the run. Up 2, right 3.
\[ m = \frac{2}{3} = \frac{\text{up } 2}{\text{right } 3} \]
Step off the slope twice to get two more points
Why: From the intercept, up 2 and right 3 lands on the next point; repeating lands on a third. Three points on one straight edge is a self-check.
\[ (0, -4) \to (3, -2) \to (6, 0) \]
Figure (svg): A coordinate grid with a line rising from the point (0, -4) through (3, -2) and crossing the horizontal axis at (6, 0), with the three points marked.
Verify the last point in the original equation
Why: Substituting the input 6 gives two thirds of 6, which is 4, minus 4, which is 0. The point where the line crosses the horizontal axis really is on the line.
\[ y = \frac{2}{3}(6) - 4 = 4 - 4 = 0 \quad \checkmark \]
Blank canvas
Draw it
Draw what Worked example: graph straight from slope-intercept form just did — the shape of it, not the line-by-line working. One picture, labels only where you need them. Then check it against the steps: anything you could not draw is a step you followed rather than understood.
Fill the middle
Fill in the blanks
From Trap: reading the intercept before solving for the output — finish the line. Write what belongs on the right of the equals sign before you look.
4(0) - 2(10) = -20 \ne 10
Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Substituting gives zero minus 20, which is negative 20, not 10.
Trap
The equation already has numbers in it, so the slope must be 4 and the vertical intercept must be 10.
\[ 4x - 2y = 10 \]
Test it: does the point with input zero and output 10 satisfy the equation?
Why: Substituting gives zero minus 20, which is negative 20, not 10. The claimed intercept is not even on the line.
\[ 4(0) - 2(10) = -20 \ne 10 \]
The slope and the intercept only sit in plain sight once the output is alone with a coefficient of one.
\[ 4x - 2y = 10 \]
Isolate the output term, then divide every term by its coefficient
Why: Subtracting the input term moves it across; dividing by negative two flips the sign of both remaining terms. That sign flip is exactly what the eyeball method misses.
\[ -2y = -4x + 10 \;\Rightarrow\; y = 2x - 5 \]
Check the real intercept
Why: Substituting an input of zero into the original equation gives negative two times the output equals 10, so the output is negative 5. That matches the solved form.
\[ m = 2, \qquad b = -5 \quad \checkmark \]
Translation
\( -2y = -4x + 10 \;\Rightarrow\; y = 2x - 5 \)
Draw it
Translate both ways. First write the expression above as a sentence with no symbols in it at all. Then cover it, and write your sentence back as notation. If the two versions disagree, the disagreement is the thing to fix.
Fill the middle
Fill in the blanks
From Worked example: solve for the output, then read the line — finish the line. Write what belongs on the right of the equals sign before you look.
-2y = -5x + 8
Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. The goal is the output alone. Subtracting the input term from both sides keeps the equation balanced and clears the left of everything but the output term.
Worked example
Find the slope and the vertical intercept of this line.
\[ 5x - 2y = 8 \]
Move the input term to the other side
Why: The goal is the output alone. Subtracting the input term from both sides keeps the equation balanced and clears the left of everything but the output term.
\[ -2y = -5x + 8 \]
Divide EVERY term by negative two
Why: Dividing must hit both terms on the right, not just the first. Two negatives divided by a negative give a positive first term and a negative constant.
\[ y = \frac{-5x}{-2} + \frac{8}{-2} = \frac{5}{2}x - 4 \]
Read the two facts off the finished form
Why: Now the coefficient of the input is genuinely the slope and the constant is genuinely the output at input zero.
\[ m = \frac{5}{2}, \qquad b = -4 \]
Verify a second point in the ORIGINAL equation
Why: The solved form predicts that an input of 2 gives an output of 1. Putting both into the original gives 10 minus 2, which is 8 - the equation we started with.
\[ 5(2) - 2(1) = 10 - 2 = 8 \quad \checkmark \]
Picture it
Animation
Shows: Each line of the worked example "solve for the output, then read the line", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: The solved form predicts that an input of 2 gives an output of 1. Putting both into the original gives 10 minus 2, which is 8 - the equation we started with.
Concept
Slope-intercept form needs the point where the input is zero. Most problems hand you some other point instead. Point-slope form is built for exactly that.
\[ y - y_1 = m(x - x_1) \]
point-slope form — The equation of the line with slope m through the known point whose coordinates are x-sub-one and y-sub-one. Both coordinates are subtracted inside the form.
It is just the slope formula with the denominator multiplied away. Nothing new is being claimed - only rearranged.
\[ m = \frac{y - y_1}{x - x_1} \;\Longrightarrow\; y - y_1 = m(x - x_1) \]
Intuition
With slope-intercept form you have to substitute a point, solve for the missing constant, and then rewrite the equation. That is three moves before you have an answer.
With point-slope form you write the answer immediately and only simplify if the problem asks for a particular form. The point you were given goes straight into the slots.
Rule of thumb: given a point and a slope, reach for point-slope. Given a graph that clearly crosses the vertical axis at a nice number, reach for slope-intercept.
Missing information
Discussion prompt
Write the equation of the line through the given point with the given slope, then put it in slope-intercept form.
What do you need to know — or decide — before the first line can be written? List everything the problem has to hand you.
Hint: Anything you would have to invent to get started is a thing the problem must supply.
Answer:
The output coordinate is negative three, and subtracting a negative three becomes adding three. That sign change is the most common slip here.
Worked example
Write the equation of the line through the given point with the given slope, then put it in slope-intercept form.
\[ (4,\, -3), \qquad m = -\frac{1}{2} \]
Drop the numbers into point-slope form exactly as given
Why: The output coordinate is negative three, and subtracting a negative three becomes adding three. That sign change is the most common slip here.
\[ y - (-3) = -\frac{1}{2}(x - 4) \;\Rightarrow\; y + 3 = -\frac{1}{2}(x - 4) \]
Distribute the slope across both terms in the parentheses
Why: Negative one half times negative four is positive two. Multiplying only the first term is the other classic slip.
\[ y + 3 = -\frac{1}{2}x + 2 \]
Subtract 3 from both sides to isolate the output
Why: This turns it into slope-intercept form so the vertical intercept becomes visible.
\[ y = -\frac{1}{2}x - 1 \]
Verify the original point lies on the finished line
Why: Substituting the input 4 gives negative two minus one, which is negative three - the output we were given. The line really does pass through the point.
\[ y = -\frac{1}{2}(4) - 1 = -2 - 1 = -3 \quad \checkmark \]
Picture it
Animation
Shows: Each line of the worked example "a point and a slope", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: Substituting the input 4 gives negative two minus one, which is negative three - the output we were given. The line really does pass through the point.
Step zero
Discussion prompt
Worked example: a line through two points — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.
Hint: It starts with: Find the slope first - nothing else can start without it
Answer:
Worked example
Write the equation of the line through both points in slope-intercept form.
\[ (-2,\, 7) \quad \text{and} \quad (4,\, -5) \]
Find the slope first - nothing else can start without it
Why: Second point minus first point on top and on the bottom, in the same order. Twelve down over six across simplifies to negative two.
\[ m = \frac{-5 - 7}{4 - (-2)} = \frac{-12}{6} = -2 \]
Feed the slope and EITHER point into point-slope form
Why: Both points are on the line, so either one produces the same final equation. Using the first point here.
\[ y - 7 = -2\bigl(x - (-2)\bigr) = -2(x + 2) \]
Distribute and solve for the output
Why: Negative two times two is negative four; adding seven to both sides then leaves the constant three.
\[ y - 7 = -2x - 4 \;\Rightarrow\; y = -2x + 3 \]
Verify with the OTHER point, the one we did not use
Why: Substituting the input 4 gives negative eight plus three, which is negative five. The unused data point lands on the line, so the equation fits both.
\[ y = -2(4) + 3 = -8 + 3 = -5 \quad \checkmark \]
Picture it
Animation
Shows: Each line of the worked example "a line through two points", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: Substituting the input 4 gives negative eight plus three, which is negative five. The unused data point lands on the line, so the equation fits both.
Concept
Standard form keeps both variables on the same side. It hides the slope, but it makes the two intercepts fall out in one step each.
\[ Ax + By = C \]
| To find | Set this to zero | Then solve for |
|---|---|---|
| the horizontal intercept | the output | the input |
| the vertical intercept | the input | the output |
The logic is simple: a point on the horizontal axis has output zero, and a point on the vertical axis has input zero. Set one to zero and the equation collapses to a one-step solve.
Hypothesis
Predict first
Worked example: graphing from standard form is about to be worked. State your hypothesis first: which rule or definition decides this one, and what is the first move it forces? Then watch whether the example agrees with you.
Correct: Set the output to zero for the horizontal intercept
Why: Every point on the horizontal axis has output zero, so the output term disappears and one division finishes it.
A hypothesis you wrote down is falsifiable; a vague sense of how it will go is not. If the example opens somewhere else, that gap is the thing worth chasing.
Worked example
Find both intercepts, then the slope, for this line.
\[ 3x + 5y = 30 \]
Set the output to zero for the horizontal intercept
Why: Every point on the horizontal axis has output zero, so the output term disappears and one division finishes it.
\[ 3x + 5(0) = 30 \;\Rightarrow\; 3x = 30 \;\Rightarrow\; x = 10 \]
Set the input to zero for the vertical intercept
Why: Same idea on the other axis: the input term disappears and the output falls out in one division.
\[ 3(0) + 5y = 30 \;\Rightarrow\; 5y = 30 \;\Rightarrow\; y = 6 \]
Use the two intercepts as the two points for the slope
Why: The intercepts are ordinary points. From the vertical intercept to the horizontal intercept the output falls 6 while the input rises 10.
\[ m = \frac{0 - 6}{10 - 0} = \frac{-6}{10} = -\frac{3}{5} \]
Rewrite in slope-intercept form as a cross-check on the slope
Why: Solving the standard form for the output should reproduce both numbers we just found, independently.
\[ 5y = -3x + 30 \;\Rightarrow\; y = -\frac{3}{5}x + 6 \]
Verify both intercepts in the original equation
Why: Thirty plus zero is 30 for the first point, and zero plus 30 is 30 for the second. Both satisfy the equation exactly as written.
\[ 3(10) + 5(0) = 30 \quad\checkmark \qquad 3(0) + 5(6) = 30 \quad\checkmark \]
Picture it
Animation
Shows: Each line of the worked example "graphing from standard form", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: Thirty plus zero is 30 for the first point, and zero plus 30 is 30 for the second. Both satisfy the equation exactly as written.
Concept
The same line can be dressed three different ways. Changing form never changes the line - it only changes which fact is easy to see.
| Form | Looks like | Best when you want |
|---|---|---|
| slope-intercept | output alone on the left | to graph fast or read the rate |
| point-slope | both coordinates subtracted | to build a line from a point and a slope |
| standard | both variables on the left | the two intercepts, or a tidy integer form |
\[ y = -\frac{3}{5}x + 6 \quad\Longleftrightarrow\quad y - 6 = -\frac{3}{5}(x - 0) \quad\Longleftrightarrow\quad 3x + 5y = 30 \]
All three of those describe the line from the last example. If a problem gives you one form and asks for another, you are being asked to do algebra, not to find a new line.
Comparison
Comparison matrix
From Three forms, one line: refill the Looks like column from what you know. The rest of the table is as it appeared.
| Form | Looks like | Best when you want |
|---|---|---|
| slope-intercept | output alone on the left | to graph fast or read the rate |
| point-slope | both coordinates subtracted | to build a line from a point and a slope |
| standard | both variables on the left | the two intercepts, or a tidy integer form |
Pattern
Whatever form you finish in, the last move is always the same: substitute a given point back and confirm both sides agree.
Prediction
Predict first
What are the slope and the vertical intercept of this line?
Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.
Correct: slope 2, intercept -4
Why: Subtract the input term to get negative three times the output equals negative six times the input plus twelve, then divide every term by negative three: the output equals two times the input minus four. So the slope is 2 and the vertical intercept is negative 4.
Check
Solve for the output on paper before you look. Watch what happens to the signs when you divide.
\[ 6x - 3y = 12 \]
Check your understanding
What are the slope and the vertical intercept of this line?
Answer: A
Why: Subtract the input term to get negative three times the output equals negative six times the input plus twelve, then divide every term by negative three: the output equals two times the input minus four. So the slope is 2 and the vertical intercept is negative 4.
Section
Part 4
Picture it
Figure (svg): Two lines with the same upward tilt drawn a fixed distance apart, and a third line crossing them at a right angle.
Discussion prompt
Read the picture before the words. What is this showing, and what is the one thing it is built to make obvious? Commit to an answer, then read on.
Hint: Name the parts, then say what changes between them — and if nothing changes, say what is being held still.
Answer:
Two lines are parallel when they never meet. Since a slope is the only thing that decides a line's direction, that can only happen one way.
Concept
Two lines are parallel when they never meet. Since a slope is the only thing that decides a line's direction, that can only happen one way.
\[ m_1 = m_2 \quad \text{and} \quad b_1 \ne b_2 \]
The second condition matters. Equal slopes and equal intercepts is not two parallel lines - it is the same line written twice.
Figure (svg): Two lines with the same upward tilt drawn a fixed distance apart, and a third line crossing them at a right angle.
Concept
Two lines are perpendicular when they cross at a right angle. Their slopes are locked together by one equation.
\[ m_1 \cdot m_2 = -1 \qquad \Longleftrightarrow \qquad m_2 = -\frac{1}{m_1} \]
negative reciprocal — Flip the fraction upside down AND change its sign. Both moves, every time - doing only one of them does not produce a right angle.
| Original slope | Perpendicular slope | Product |
|---|---|---|
| 3 | negative one third | -1 |
| two fifths | negative five halves | -1 |
| negative four | one fourth | -1 |
| zero (horizontal) | undefined (vertical) | no product - special case |
Matching
Match the pairs
Match each term to the definition this lesson gave it — not the one you would guess from the word.
Why: These are the working definitions of linear function, slope, slope-intercept form, point-slope form, negative reciprocal as Linear Functions and Modeling uses them. Pairing them correctly is the test of whether you could state each one with the slide switched off.
Intuition
Draw the slope triangle for a line: over 3, up 2. Now rotate that whole triangle a quarter turn, the way you would turn a piece of paper.
The leg that pointed across now points up, and the leg that pointed up now points across. So the rise and run trade places - that is the flip.
But the rotation also swings one leg into the opposite direction, so one of the two becomes negative. That is the sign change. Rotating gives you both moves at once, which is why you can never take just one.
The horizontal and vertical pair is the same story at its extreme: a flat line rotated a quarter turn is a vertical line, which is why zero slope pairs with undefined slope.
Ranking
Put in order
Put the moves of Worked example: a parallel line through a point into the order they have to happen.
Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Standard form hides the slope. Subtracting the input term and dividing every term by 3 puts it in slope-intercept form.
Worked example
Write the equation of the line through the given point that is parallel to the given line.
\[ (6,\, -1), \qquad 2x + 3y = 12 \]
Solve the given line for its output to expose the slope
Why: Standard form hides the slope. Subtracting the input term and dividing every term by 3 puts it in slope-intercept form.
\[ 3y = -2x + 12 \;\Rightarrow\; y = -\frac{2}{3}x + 4 \]
Copy the slope unchanged
Why: Parallel means identical direction, so the new line uses the very same slope. Only the intercept will differ.
\[ m = -\frac{2}{3} \]
Use point-slope form with the given point
Why: The point has input 6 and output negative 1, so the output side becomes a plus one and the input side becomes a minus six.
\[ y + 1 = -\frac{2}{3}(x - 6) \]
Distribute and isolate the output
Why: Negative two thirds times negative six is positive four; subtracting one from both sides leaves the constant three.
\[ y + 1 = -\frac{2}{3}x + 4 \;\Rightarrow\; y = -\frac{2}{3}x + 3 \]
Verify the point is on the new line and the slopes still match
Why: Substituting the input 6 gives negative four plus three, which is negative one - the point checks. Both slopes are negative two thirds and the intercepts differ, so the lines are parallel and not identical.
\[ -\frac{2}{3}(6) + 3 = -4 + 3 = -1 \quad \checkmark \]
Picture it
Animation
Shows: Each line of the worked example "a parallel line through a point", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: Substituting the input 6 gives negative four plus three, which is negative one - the point checks. Both slopes are negative two thirds and the intercepts differ, so the lines are parallel and not identical.
Estimation
Predict first
Same point, same given line - but now the new line must cross it at a right angle.
Commit before you compute: what does Worked example: a perpendicular line through the same point come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.
Correct: Verify the point satisfies the finished equation
Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Subtracting one from both sides gives the slope-intercept form, and substituting the input 6 gives nine minus ten, which is negative one - the point we were handed.
Worked example
Same point, same given line - but now the new line must cross it at a right angle.
\[ (6,\, -1), \qquad 2x + 3y = 12 \]
Start from the slope we already found
Why: The given line is unchanged, so its slope is still negative two thirds. Only what we do with it changes.
\[ m_{\text{given}} = -\frac{2}{3} \]
Flip the fraction and change its sign
Why: Flipping negative two thirds gives negative three halves; changing the sign gives positive three halves. Both moves, in either order.
\[ m_{\perp} = -\frac{1}{-2/3} = \frac{3}{2} \]
Confirm the product is negative one before going further
Why: This is a two-second insurance policy against the half-a-reciprocal mistake. The threes cancel and the twos cancel, leaving negative one.
\[ \left(-\frac{2}{3}\right)\left(\frac{3}{2}\right) = -\frac{6}{6} = -1 \quad \checkmark \]
Use point-slope form with the given point
Why: Same point as before, so the same slots get filled - only the slope has changed.
\[ y + 1 = \frac{3}{2}(x - 6) = \frac{3}{2}x - 9 \]
Verify the point satisfies the finished equation
Why: Subtracting one from both sides gives the slope-intercept form, and substituting the input 6 gives nine minus ten, which is negative one - the point we were handed.
\[ y = \frac{3}{2}x - 10, \qquad \frac{3}{2}(6) - 10 = 9 - 10 = -1 \quad \checkmark \]
Picture it
Animation
Shows: Each line of the worked example "a perpendicular line through the same point", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: Subtracting one from both sides gives the slope-intercept form, and substituting the input 6 gives nine minus ten, which is negative one - the point we were handed.
Anomaly
Predict first
A student writes this, and it looks reasonable:
The line has slope four fifths, so a perpendicular line just needs the opposite sign.
It is wrong. Say what breaks — and say it before you turn the page.
Correct: The product comes out to negative sixteen twenty-fifths, not negative one.
Do both moves and then prove it with the product.
Why: The product comes out to negative sixteen twenty-fifths, not negative one. These two lines are mirror images of each other, not perpendicular.
Trap
The line has slope four fifths, so a perpendicular line just needs the opposite sign.
\[ m_1 = \frac{4}{5} \quad\Rightarrow\quad m_2 \stackrel{?}{=} -\frac{4}{5} \]
Test it with the product rule
Why: The product comes out to negative sixteen twenty-fifths, not negative one. These two lines are mirror images of each other, not perpendicular.
\[ \frac{4}{5} \cdot \left(-\frac{4}{5}\right) = -\frac{16}{25} \ne -1 \]
Flipping only is just as wrong
Why: Five fourths times four fifths is positive one, not negative one. One move alone never gets there - it takes both.
Do both moves and then prove it with the product.
\[ m_1 = \frac{4}{5} \quad\Rightarrow\quad m_2 = -\frac{5}{4} \]
Flip, then negate
Why: Four fifths turned upside down is five fourths; changing the sign gives negative five fourths. Say the two words out loud as you do it.
Check the product is exactly negative one
Why: The fours cancel and the fives cancel, leaving negative one. That single line of arithmetic catches every version of this mistake.
\[ \frac{4}{5} \cdot \left(-\frac{5}{4}\right) = -\frac{20}{20} = -1 \quad \checkmark \]
Break the constraint
Discussion prompt
The rule this trap just fixed:
Four fifths turned upside down is five fourths; changing the sign gives negative five fourths. Say the two words out loud as you do it.
Now break it on purpose. Build a case that violates it and follow the consequences until something visibly fails. Where does the failure first show up — and would you have noticed it if you had not been looking?
Hint: The dangerous rules are the ones whose violation still produces an answer. If yours fails loudly, try to find one that fails quietly.
Answer:
The product comes out to negative sixteen twenty-fifths, not negative one. These two lines are mirror images of each other, not perpendicular.
Constraint
Discussion prompt
Run Pattern: parallel or perpendicular through a point with this step confiscated:
For parallel: keep that slope exactly. For perpendicular: flip it and change its sign.
Is it still possible? If it is, say what takes its place and what it costs you. If it is not, say exactly what that step was providing that nothing else does.
Hint: A step you can drop for free was never load-bearing. If you cannot drop it, name the thing that goes wrong the moment it is gone.
Answer:
Pattern
Special cases to keep in your pocket: perpendicular to a horizontal line is a vertical line, and perpendicular to a vertical line is a horizontal line. No fractions involved.
Edge cases
Discussion prompt
Pattern: parallel or perpendicular through a point works on the cases you have just seen. Push it to the edge: what is the most degenerate input it still handles — empty, zero, one item, everything equal — and what is the first case where it stops being true? Name the case, not just "it breaks".
Hint: Try the smallest legal input, then the largest, then the one where two things collide. Methods are specified at their edges; the middle takes care of itself.
Answer:
Special cases to keep in your pocket: perpendicular to a horizontal line is a vertical line, and perpendicular to a vertical line is a horizontal line. No fractions involved.
Picture it
Animation
Shows: Parallel and perpendicular slopes — a rendered Manim animation.
Rendered with Manim.
Takeaway: Parallel shares the slope; perpendicular uses the negative reciprocal.
Commit first
Predict first
Which equation describes that perpendicular line?
Commit to an answer, then rate it — certain, fairly sure, or guessing — and write the rating down before you turn the page.
Correct: y = 2x + 11
Why: The negative reciprocal of negative one half is positive two. Point-slope gives the output minus three equals two times the input plus four, so the output equals two times the input plus eleven. Check: two times negative four plus eleven is negative eight plus eleven, which is three.
The rating matters as much as the answer: confident-and-wrong is the combination that survives revision, because nothing about it feels like it needs revisiting.
Check
Find the perpendicular slope first, then run it through point-slope form with the given point.
\[ \text{Through } (-4,\, 3), \text{ perpendicular to } y = -\frac{1}{2}x + 7 \]
Check your understanding
Which equation describes that perpendicular line?
Answer: A
Why: The negative reciprocal of negative one half is positive two. Point-slope gives the output minus three equals two times the input plus four, so the output equals two times the input plus eleven. Check: two times negative four plus eleven is negative eight plus eleven, which is three.
Section
Part 5
Concept
A linear model is the same equation you have been writing all deck, plus two sentences saying what each letter measures and in what units.
linear model — A linear function used to describe a real quantity, together with a stated meaning and unit for the input and for the output.
| Piece | Meaning in context | Units |
|---|---|---|
| input | what you control or measure along | hours, months, miles |
| output | what you are predicting | dollars, degrees, gallons |
| slope | how much output per one unit of input | output units per input unit |
| intercept | the output before the input starts running | output units |
Without units the model is just symbols. With units, the slope becomes a sentence you can say out loud - and saying it is how you catch a wrong model.
Trade off
Comparison matrix
From A model is a line with units attached: every row here is a choice with a cost. Fill the Units column, then say which row you would actually pick and what you give up for it.
| Piece | Meaning in context | Units |
|---|---|---|
| input | what you control or measure along | hours, months, miles |
| output | what you are predicting | dollars, degrees, gallons |
| slope | how much output per one unit of input | output units per input unit |
| intercept | the output before the input starts running | output units |
Intuition
Read the problem hunting for two things and nothing else: a one-time amount and a repeating amount.
| Words that signal the intercept | Words that signal the slope |
|---|---|
| starting, initial, sign-up fee, deposit | per, each, every, rate, monthly, hourly |
| at the beginning, before, flat fee | increases by, decreases by, loses, gains |
The one-time amount is the intercept. The repeating amount is the slope, and if the quantity is shrinking it carries a minus sign. Everything else in the paragraph is decoration.
Comparison
Comparison matrix
From Every word problem is hiding the same two numbers: refill the Words that signal the slope column from what you know. The rest of the table is as it appeared.
| Words that signal the intercept | Words that signal the slope |
|---|---|
| starting, initial, sign-up fee, deposit | per, each, every, rate, monthly, hourly |
| at the beginning, before, flat fee | increases by, decreases by, loses, gains |
Step zero
Discussion prompt
Worked example: a gym membership model — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.
Hint: It starts with: Define the variables and their units first
Answer:
Worked example
A gym charges a one-time joining fee of 75 dollars plus 28 dollars each month. Write a model for the total cost, then find the cost after 18 months.
Define the variables and their units first
Why: Naming the quantities before writing symbols is what makes the answer interpretable - and it forces you to notice which one is the input.
\[ t = \text{months of membership}, \qquad C(t) = \text{total cost in dollars} \]
The joining fee is the intercept
Why: It is paid once, at the moment when zero months have passed, so it is the output when the input is zero.
\[ b = 75 \ \text{dollars} \]
The monthly charge is the slope, with units
Why: The word each signals a repeating amount: 28 dollars of output for every one month of input. The cost rises, so the slope is positive.
\[ m = 28 \ \frac{\text{dollars}}{\text{month}} \]
Assemble the model
Why: Slope times input plus intercept. The units work out: dollars per month times months gives dollars, plus dollars gives dollars.
\[ C(t) = 28t + 75 \]
Evaluate at 18 months
Why: Twenty-eight times eighteen is 504, and adding the 75 dollar fee gives 579 dollars.
\[ C(18) = 28(18) + 75 = 504 + 75 = 579 \]
Verify the model against the story at two inputs
Why: At zero months the model gives 75 dollars, which is exactly the joining fee before any monthly charge. And from month 17 to month 18 the cost goes from 551 to 579, a rise of 28 dollars - the stated monthly rate.
\[ C(0) = 75 \quad\checkmark \qquad C(18) - C(17) = 579 - 551 = 28 \quad\checkmark \]
Picture it
Animation
Shows: Each line of the worked example "a gym membership model", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: At zero months the model gives 75 dollars, which is exactly the joining fee before any monthly charge. And from month 17 to month 18 the cost goes from 551 to 579, a rise of 28 dollars - the stated monthly rate.
Fill the middle
Fill in the blanks
From Worked example: a model from two data points — finish the line. Write what belongs on the right of the equals sign before you look.
C(h) = 40h + 65
Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Hours is what varies and cost is what depends on it, so hours is the input.
Worked example
A plumber charges 145 dollars for a 2-hour job and 265 dollars for a 5-hour job, and the total is linear in the hours. Find the model, the hourly rate, and the service-call fee.
Turn the two sentences into two points
Why: Hours is what varies and cost is what depends on it, so hours is the input. Each sentence becomes an ordered pair.
\[ (2,\, 145) \quad \text{and} \quad (5,\, 265) \]
Compute the slope, carrying units
Why: The cost rose 120 dollars while the time rose 3 hours, so the rate is 40 dollars for every hour of work. That is the hourly rate the problem asked for.
\[ m = \frac{265 - 145}{5 - 2} = \frac{120 \ \text{dollars}}{3 \ \text{hours}} = 40 \ \frac{\text{dollars}}{\text{hour}} \]
Substitute one point to find the intercept
Why: Using the 2-hour job: forty times two is eighty, so the remaining 65 dollars is charged no matter how long the job takes.
\[ 145 = 40(2) + b \;\Rightarrow\; 145 = 80 + b \;\Rightarrow\; b = 65 \]
Write the model and say what each number means
Why: The intercept of 65 dollars is a flat service-call fee - what you pay for the plumber showing up at all - and the slope of 40 dollars per hour is the labor rate.
\[ C(h) = 40h + 65 \]
Verify with the OTHER data point
Why: The 5-hour job was never used to find the intercept, so it is a genuine test. Forty times five is 200, plus 65 gives 265 dollars - exactly the quoted price.
\[ C(5) = 40(5) + 65 = 200 + 65 = 265 \quad \checkmark \]
Concept
A model is trustworthy where the data is, and increasingly speculative where it is not.
interpolation — Predicting at an input that falls between the inputs you actually observed. Generally reliable, because the data brackets your prediction.
extrapolation — Predicting at an input outside the observed range. Riskier the farther you go, because nothing you measured supports it.
Real quantities stop being linear eventually. A candle burns out, a car cannot be worth a negative amount, and a plumber will not work a hundred-hour shift. Always ask whether the answer is physically possible.
Picture it
Animation
Shows: Interpolate freely, extrapolate carefully — a rendered Manim animation.
Rendered with Manim.
Takeaway: A linear model of growth eventually predicts something absurd.
Missing information
Discussion prompt
Use the model from the last example to price a 4-hour job and a 40-hour job, and say how much you trust each.
What do you need to know — or decide — before the first line can be written? List everything the problem has to hand you.
Hint: Anything you would have to invent to get started is a thing the problem must supply.
Answer:
Forty times four is 160, plus the 65 dollar service fee gives 225 dollars.
Worked example
Use the model from the last example to price a 4-hour job and a 40-hour job, and say how much you trust each.
\[ C(h) = 40h + 65 \quad \text{built from data at } h = 2 \text{ and } h = 5 \]
Predict the 4-hour job
Why: Forty times four is 160, plus the 65 dollar service fee gives 225 dollars.
\[ C(4) = 40(4) + 65 = 160 + 65 = 225 \]
Label it interpolation and trust it
Why: Four hours sits between the two observed jobs of 2 and 5 hours, so the data brackets the prediction. This is the safe kind of estimate.
Predict the 40-hour job
Why: Forty times forty is 1600, plus 65 gives 1665 dollars.
\[ C(40) = 40(40) + 65 = 1600 + 65 = 1665 \]
Label it extrapolation and flag it
Why: Forty hours is eight times the longest job we measured. A real plumber would quote a multi-day project differently, so the arithmetic is right but the model may not apply.
Verify the model still reproduces the original data
Why: Before trusting any prediction, re-check a known point: forty times two plus 65 is 145 dollars, the actual quoted price for the 2-hour job. The model is intact.
\[ C(2) = 40(2) + 65 = 80 + 65 = 145 \quad \checkmark \]
Picture it
Animation
Shows: Each line of the worked example "predicting with the plumber model", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: Before trusting any prediction, re-check a known point: forty times two plus 65 is 145 dollars, the actual quoted price for the 2-hour job. The model is intact.
Anomaly
Predict first
A student writes this, and it looks reasonable:
Using the plumber model, a student is asked what the slope means in context and answers: the job costs 40 dollars.
It is wrong. Say what breaks — and say it before you turn the page.
Correct: The 2-hour job cost 145 dollars, not 40.
Read the slope as a fraction of units, then turn that fraction into an English sentence.
Why: The 2-hour job cost 145 dollars, not 40. The number 40 is not a price at all - it never stands alone as a total.
Trap
Using the plumber model, a student is asked what the slope means in context and answers: the job costs 40 dollars.
\[ C(h) = 40h + 65 \]
Test the claim against the data
Why: The 2-hour job cost 145 dollars, not 40. The number 40 is not a price at all - it never stands alone as a total.
\[ C(2) = 145 \ne 40 \]
The units were dropped, so the meaning went with them
Why: A slope is always a ratio of two units. Reporting it as a bare amount of money throws away the per-hour half of what it says.
Read the slope as a fraction of units, then turn that fraction into an English sentence.
\[ m = 40 \ \frac{\text{dollars}}{\text{hour}} \]
Say it as: for each additional hour, the cost increases by 40 dollars
Why: The words for each additional carry the per, and the words increases by carry the sign. A negative slope would read decreases by.
Check the sentence against two rows of the data
Why: Going from the 2-hour job to the 5-hour job is 3 more hours and 120 more dollars, and 120 divided by 3 is 40 dollars per hour. The sentence matches the data exactly.
\[ \frac{265 - 145}{5 - 2} = 40 \quad \checkmark \]
Notation
Annotate
From Trap: reporting the slope as a total instead of a rate — read this one piece at a time. What is each part doing?
On: \( \frac{265 - 145}{5 - 2} = 40 \quad \checkmark \)
Estimation
Predict first
A delivery van is bought for 32000 dollars and is worth 20600 dollars four years later. Assuming the value falls linearly, model it, predict the value after 7 years, and find when it hits zero.
Commit before you compute: what does Worked example: straight-line depreciation come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.
Correct: Verify the model reproduces the four-year value
Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Negative 2850 times 4 is negative 11400, and 32000 minus 11400 is 20600 dollars - exactly the given value at year four.
Worked example
A delivery van is bought for 32000 dollars and is worth 20600 dollars four years later. Assuming the value falls linearly, model it, predict the value after 7 years, and find when it hits zero.
Set up the variables with the purchase as time zero
Why: Choosing the purchase moment as the zero input makes the purchase price the intercept for free, which saves an algebra step.
\[ t = \text{years since purchase}, \qquad V(t) = \text{value in dollars}, \qquad b = 32000 \]
Compute the slope from the two known values
Why: The value fell 11400 dollars over 4 years. A falling quantity gives a negative slope, and 11400 divided by 4 is 2850.
\[ m = \frac{20600 - 32000}{4 - 0} = \frac{-11400}{4} = -2850 \ \frac{\text{dollars}}{\text{year}} \]
Write the model and read the slope aloud
Why: The van loses 2850 dollars of value each year. Saying it in words is the fastest check that the sign is right - a used van should be worth less, not more.
\[ V(t) = -2850t + 32000 \]
Predict the value after 7 years
Why: Negative 2850 times 7 is negative 19950, and 32000 minus 19950 is 12050 dollars.
\[ V(7) = -2850(7) + 32000 = -19950 + 32000 = 12050 \]
Find when the model reaches zero
Why: Set the value to zero and solve. Thirty-two thousand divided by 2850 is about 11.2 years - and beyond that the model would claim a negative value, which is where it stops making sense.
\[ 0 = -2850t + 32000 \;\Rightarrow\; t = \frac{32000}{2850} \approx 11.2 \ \text{years} \]
Verify the model reproduces the four-year value
Why: Negative 2850 times 4 is negative 11400, and 32000 minus 11400 is 20600 dollars - exactly the given value at year four.
\[ V(4) = -2850(4) + 32000 = -11400 + 32000 = 20600 \quad \checkmark \]
Pattern
Finish by asking two questions: does the model reproduce the data I was given, and is the prediction inside the range where the data lives?
Picture it
Animation
Shows: Reading a model back into words — a rendered Manim animation.
Rendered with Manim.
Takeaway: The slope is the variable cost; the intercept is the fixed cost.
Prediction
Predict first
Which model describes the gallons remaining after t minutes?
Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.
Correct: V(t) = 50 - 3t
Why: The 50 gallons are there before any draining, so 50 is the intercept. The tank loses 3 gallons for every minute, so the slope is negative 3 gallons per minute. After 10 minutes the model gives 50 minus 30, which is 20 gallons remaining.
Check
A tank holds 50 gallons of water and drains steadily at 3 gallons per minute. Let the input be minutes since the drain opened and the output be gallons remaining.
Check your understanding
Which model describes the gallons remaining after t minutes?
Answer: A
Why: The 50 gallons are there before any draining, so 50 is the intercept. The tank loses 3 gallons for every minute, so the slope is negative 3 gallons per minute. After 10 minutes the model gives 50 minus 30, which is 20 gallons remaining.
Section
Part 6
Picture it
Figure (svg): A scatter plot of eleven dots drifting upward from lower left to upper right, with a dashed straight line drawn through the middle of the cloud.
Discussion prompt
Read the picture before the words. What is this showing, and what is the one thing it is built to make obvious? Commit to an answer, then read on.
Hint: Name the parts, then say what changes between them — and if nothing changes, say what is being held still.
Answer:
Every model so far came from perfect numbers. Measured data never behaves that well - it clusters near a line instead of landing on one.
Concept
Every model so far came from perfect numbers. Measured data never behaves that well - it clusters near a line instead of landing on one.
scatter plot — A graph of paired measurements, one dot per observation, with the explanatory quantity on the horizontal axis and the response on the vertical axis.
Figure (svg): A scatter plot of eleven dots drifting upward from lower left to upper right, with a dashed straight line drawn through the middle of the cloud.
The question is no longer which line goes through the points - none does. It is which line comes closest to all of them at once.
Explain it
Discussion prompt
Explain Real data does not sit on a line to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.
Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.
Answer:
Every model so far came from perfect numbers. Measured data never behaves that well - it clusters near a line instead of landing on one.
Intuition
Imagine laying a ruler on the scatter plot and rotating it until the dots are as evenly balanced above and below it as you can get them. That ruler position is the line of best fit.
line of best fit — The line that makes the total vertical distance from the data points to the line as small as possible. A calculator or spreadsheet finds it; you interpret it.
You will not compute one by hand in this course. What you will be asked to do is read its slope and intercept in context, and use it to predict - exactly the skills from Part 5.
One warning built into the picture: the best-fit line is a summary, not a promise. Individual dots sit above and below it, so a prediction is a typical value, not a guarantee for any one case.
Analogy
Discussion prompt
Explain The line of best fit by analogy to something with no College Algebra in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.
Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.
Answer:
You will not compute one by hand in this course. What you will be asked to do is read its slope and intercept in context, and use it to predict - exactly the skills from Part 5.
Picture it
Animation
Shows: A line of best fit — a rendered Manim animation.
Rendered with Manim.
Takeaway: No line passes through every point; the best one minimises the misses.
Concept
Correlation describes how tightly the dots hug a straight line, and which way they lean.
| What you see | Direction | Strength |
|---|---|---|
| dots rise together, tightly packed | positive | strong |
| dots fall together, tightly packed | negative | strong |
| dots lean but scatter widely | positive or negative | weak |
| dots form a shapeless cloud | none | none |
Direction always matches the sign of the best-fit slope: a positive slope means the two quantities tend to increase together, a negative slope means one tends to fall as the other rises.
Strength is a separate question from steepness. A very steep line through a very loose cloud is still a weak relationship.
Discrimination
Sort into buckets
Sort these by Strength, from memory, without looking back at Correlation: direction and strength. Telling them apart on the spot is the skill; the table is only where the answer happens to be written down.
Anomaly
Predict first
A student writes this, and it looks reasonable:
Across a city's summer days, ice-cream sales and swimming-pool accidents are strongly positively correlated. So buying ice cream must cause pool accidents.
It is wrong. Say what breaks — and say it before you turn the page.
Correct: A scatter plot records that two quantities moved together.
State what a correlation actually licenses you to say, and stop there.
Why: A scatter plot records that two quantities moved together. It has no way to record which one, if either, made the other happen.
Trap
Across a city's summer days, ice-cream sales and swimming-pool accidents are strongly positively correlated. So buying ice cream must cause pool accidents.
The data cannot support that sentence
Why: A scatter plot records that two quantities moved together. It has no way to record which one, if either, made the other happen.
A third quantity explains both
Why: Hot days raise ice-cream sales and also send more people to pools. Temperature drives both columns, so they rise together without touching each other.
State what a correlation actually licenses you to say, and stop there.
Use the phrase is associated with, not causes
Why: Days with higher ice-cream sales tend to be days with more pool accidents. That sentence is fully supported by the data and claims nothing extra.
Check for a lurking third quantity before believing any causal story
Why: Ask what else could move both columns at once. If you find a plausible candidate - here, temperature - the causal claim is unsupported.
Two truths and a lie
Sort into buckets
Some of these hold up and some are the exact mistakes this lesson is built to prevent. Sort them.
Step zero
Discussion prompt
Worked example: using a best-fit line — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.
Hint: It starts with: Find the slope from the two points on the fitted line
Answer:
Worked example
A study records weekly study hours and exam scores for a class. The best-fit line passes through the two points below. Find its equation, interpret the slope, and predict the score for 6 hours of study.
\[ (2,\, 71) \quad \text{and} \quad (8,\, 98) \]
Find the slope from the two points on the fitted line
Why: These points are on the line itself, not raw data, so the ordinary slope formula applies. Twenty-seven points of score over 6 hours of study.
\[ m = \frac{98 - 71}{8 - 2} = \frac{27}{6} = 4.5 \ \frac{\text{points}}{\text{hour}} \]
Substitute one point to find the intercept
Why: Using the first point: 4.5 times 2 is 9, so the intercept is 71 minus 9, which is 62 points.
\[ 71 = 4.5(2) + b \;\Rightarrow\; 71 = 9 + b \;\Rightarrow\; b = 62 \]
Write the model and interpret both numbers carefully
Why: Each additional hour of weekly study is associated with about 4.5 more exam points, and a student who studies zero hours is predicted to score about 62. Associated with, not causes.
\[ S(h) = 4.5h + 62 \]
Predict the score for 6 hours
Why: Four and a half times six is 27, plus 62 gives 89 points. Six hours sits between the two fitted points, so this is interpolation and reasonably safe.
\[ S(6) = 4.5(6) + 62 = 27 + 62 = 89 \]
Verify the equation reproduces the second fitted point
Why: Four and a half times eight is 36, and 36 plus 62 is 98 - the second point we were given and never used. The equation of the line is right.
\[ S(8) = 4.5(8) + 62 = 36 + 62 = 98 \quad \checkmark \]
Check
Use the study model from the last example. Read every choice carefully - three of them are true-sounding sentences that the data does not support.
\[ S(h) = 4.5h + 62 \]
Check your understanding
What does the slope of 4.5 mean here?
Answer: A
Why: The slope is a rate of output units per input unit: 4.5 points per hour of study. Going from 2 hours to 3 hours moves the predicted score from 71 to 75.5, a rise of 4.5 points, and the wording stays associational because this is observational data.
Concept
Put two linear models on the same axes and the crossing point answers a question you meet constantly: when are they equal?
Figure (svg): Two straight lines drawn on a pair of axes crossing at a single marked point labeled (2, 3).
solution of a linear system — An ordered pair that satisfies both equations at once. Graphically it is the point where the two lines intersect.
Two lines can cross once, never (parallel, different intercepts), or everywhere (the same line twice). Those three pictures are the three possible answers, and you already know how to tell them apart from the slopes.
Counterexample
Discussion prompt
Put two linear models on the same axes and the crossing point answers a question you meet constantly: when are they equal?
That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.
Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.
Ranking
Put in order
Put the moves of Worked example: where do two lines meet? into the order they have to happen.
Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. The slopes are 2 and negative 1, which are different, so the lines are not parallel.
Worked example
Find the point that lies on both lines.
\[ y = 2x - 1 \qquad \text{and} \qquad y = -x + 5 \]
Check the slopes first to predict the answer type
Why: The slopes are 2 and negative 1, which are different, so the lines are not parallel. Exactly one crossing point must exist.
Set the two outputs equal to each other
Why: At the crossing point the two lines produce the same output for the same input, so their right-hand sides must be equal there.
\[ 2x - 1 = -x + 5 \]
Collect the variable on one side and solve
Why: Adding the input term to both sides gives three of them on the left; adding one to both sides gives six on the right; dividing by three gives an input of two.
\[ 3x = 6 \;\Rightarrow\; x = 2 \]
Substitute back to get the output
Why: Using the first equation: two times two minus one is three. The crossing point is the pair with input two and output three.
\[ y = 2(2) - 1 = 3 \;\Rightarrow\; (2,\, 3) \]
Verify the point in BOTH original equations
Why: The first gives four minus one, which is three. The second gives negative two plus five, which is also three. A system answer is only checked when both equations agree - checking one proves nothing.
\[ 2(2) - 1 = 3 \quad\checkmark \qquad -(2) + 5 = 3 \quad\checkmark \]
Picture it
Animation
Shows: Each line of the worked example "where do two lines meet?", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: The slopes are 2 and negative 1, which are different, so the lines are not parallel. Exactly one crossing point must exist.
Elimination
Eliminate the wrong options
Which equation describes the line through the point with input 5 and output -2 that is parallel to the horizontal axis?
3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.
Survives elimination: A
Why: A line parallel to the horizontal axis is horizontal, so its slope is zero and every point on it has the same output. The given point has output negative two, so the output is locked at negative two while the input is free.
Check
Picture it before you compute. A line parallel to the horizontal axis is flat, so ask which coordinate is being held fixed.
Check your understanding
Which equation describes the line through the point with input 5 and output -2 that is parallel to the horizontal axis?
Answer: A
Why: A line parallel to the horizontal axis is horizontal, so its slope is zero and every point on it has the same output. The given point has output negative two, so the output is locked at negative two while the input is free.
Ranking
Put in order
These are the steps of Pattern: the whole deck on one card, scrambled. Put them back in order before the next slide shows you.
Why: This is the order the recipe itself gives. Recalling the sequence without the slide in front of you is the difference between recognising the method and being able to run it — most of what goes wrong in practice is a step done out of turn.
Pattern
Real world
Discussion prompt
Outside this lesson: where does Linear Functions and Modeling actually turn up? Name one concrete situation — a job, a piece of software someone ships, a decision somebody has to make — and say which part of Pattern: the whole deck on one card is doing the work in it.
Hint: Vague is the failure mode here. "Engineering" is not a situation; "deciding whether this build is fast enough to ship" is.
Answer:
That deck covers everything linear. It shows a constant rate of change in a table, in a graph, and in an equation, finds slope from two points and reads it as a rate with units, and works through slope-intercept, point-slope, and standard form, along with horizontal and vertical lines and the slopes of parallel and perpendicular lines. It then builds a model from a description or from two data points and predicts with it, and closes with scatter plots, best-fit lines, and correlation. It targets four classic errors: subtracting the coordinates in mismatched order, confusing zero slope with undefined slope, taking only half of a negative reciprocal, and reading the y-intercept off an equation that has not yet been solved for y.
Connect it up
Draw it
One page, no notation unless you need it: draw how these connect — What Makes a Function Linear · Slope: The Number That Runs Everything · Every Form of a Line · Parallel and Perpendicular · Building a Model That Predicts · Real Data: Scatter Plots and Fit. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.
Recap
One constant rate of change generated this entire deck. Slope is that rate, the forms are three ways of writing it down, parallel and perpendicular are two rules about it, and a model is that rate with units attached.
| If you are given | Reach for |
|---|---|
| two points | the slope formula, then point-slope form |
| a point and a slope | point-slope form |
| standard form | solve for the output, or set each variable to zero for the intercepts |
| parallel or perpendicular | the same slope, or the flipped and negated slope |
| a word problem | the one-time amount and the per-unit rate |
Next up: quadratic functions, where the rate of change stops being constant - and everything you just learned about slope becomes the tool for measuring how it changes.
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