Defines a linear function by the property that its rate of change is constant, which is exactly what makes its graph straight. Computes slope from two points, from a table and from a description, interprets its sign and its units, and introduces the slope-intercept and point-slope forms along with the special cases of horizontal and vertical lines.
Subject: Precalculus · 65 slides · symbolic lesson
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Title
Precalculus · Chapter 2 — Linear Functions
§2.1 Linear Functions, pp. 182-204
Objectives
Five things, each one you can check yourself on paper.
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 182-204 — the pages these objectives are drawn from
Warm-up
Everyone can recognise a straight line. The question is what property of the rule produces one.
Discussion prompt
A function's average rate of change from 0 to 1 is 3, from 1 to 2 is 3, and from 0 to 100 is also 3. What does that force the graph to look like, and why?
Hint: What would have to happen somewhere for the graph to bend?
Answer:
It forces the graph to be straight. Bending would mean the steepness differs somewhere, and a different steepness on some stretch would show up as a different average rate of change over an interval containing it.
So the rate being the same over every interval is not a consequence of straightness — it is the same fact stated algebraically. That is the definition this section runs on.
It is worth noticing that this is the property §1.3 could not find for a general function. A parabola's average rate of change depends on which interval you pick; a line's does not, and that single difference is why lines are the family everything else is measured against.
Concept
A function is linear when its rate of change is constant: the same over every interval, however long or short. That constant is the slope, and it is what makes the graph a straight line.
linear function — A function whose rate of change is constant. Its graph is a straight line, and its rule can be written as a constant multiple of the input plus a constant.
\[ f(x)=mx+b, \qquad m = \frac{\Delta y}{\Delta x} \text{ is the same for every interval} \]
The constant b is the output at the input zero, so it is where the graph meets the vertical axis. The constant m is the rate. Between them they determine the line completely, which is why two numbers are enough to describe an object that contains infinitely many points.
Figure (svg): A straight line with three slope triangles of different widths drawn beneath it, each labelled with its rise and run, showing that the ratio is the same for all three
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 182-186
Section
Section 1
Concept
For a linear function, the average rate of change computed between any two points comes out the same. That common value is the slope.
\[ m = \frac{y_2-y_1}{x_2-x_1} \]
That last bullet is the practical test for whether data is linear. If the inputs advance by a constant amount and the outputs also advance by a constant amount, the relationship is linear. If the input steps are unequal, divide each output step by its own input step and check the quotients agree.
Figure (svg): A straight line with three slope triangles of different widths drawn beneath it, each labelled with its rise and run, showing that the ratio is the same for all three
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 182-190
Picture it
The triangles have different sizes and the same shape, which is what equal ratios mean geometrically.
Figure (svg): A straight line with three slope triangles of different widths drawn beneath it, each labelled with its rise and run, showing that the ratio is the same for all three
Any two points on the line give the same slope. That is why a line needs no interval specified when its rate is quoted, and a curve always does.
Worked example
Subtract in matching order and divide.
\[ \text{Find the slope of the line through } (2,-3) \text{ and } (6,5). \]
Subtract the outputs
Why: Second minus first.
\[ 5 - (-3) = 8 \]
Subtract the inputs in the same order
Why: Second minus first again.
\[ 6 - 2 = 4 \]
Divide
Why: Rise over run.
\[ \frac{8}{4} \]
Simplify
Why: The slope is a single number.
\[ m = 2 \]
Figure (svg): Two plotted points joined by a line, with the slope formula written beneath and the rise and run marked on the diagram as the differences of the coordinates
\[ m = \frac{5-(-3)}{6-2} = 2 \]
Verify: reverse the order and confirm
Why: Taking the points the other way gives negative 8 over negative 4, which is also 2. Both signs flipped, so the quotient did not — which is why either order is acceptable as long as it is used consistently in both the numerator and the denominator.
\[ \frac{-3 - 5}{2 - 6} = -8 / - 4 = 2 \]
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 184-186
Sorting
Check whether equal input steps give equal output steps.
Sort into buckets
Sort each table or rule.
Worked example
Constant input steps and constant output steps is the test.
\[ \begin{array}{c|cccc} x & 1 & 3 & 5 & 7 \\ \hline y & 4 & 10 & 16 & 22 \end{array} \]
Check the input steps
Why: Each is the same.
\[ \text{steps of } 2 \]
Check the output steps
Why: Each is the same as well.
\[ \text{steps of } 6 \]
Divide one by the other
Why: Output step over input step.
\[ \frac{6}{2} = 3 \]
Conclude
Why: Constant rate, so linear with slope 3.
Figure (svg): The solution to Worked example decide whether a table is linear shown as a ladder of expressions, one row per legal move
\[ \text{Linear, } m=3. \]
Verify: confirm with a non-adjacent pair
Why: From the first point to the last: the outputs differ by 18 and the inputs by 6, giving 3 again. Testing a wide interval as well as the narrow ones is worth doing, because a table can have equal steps between neighbours by coincidence over a short stretch.
\[ \frac{22 - 4}{7 - 1} = \frac{18}{6} = 3 \]
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 187-189
Trap
\[ m = \frac{5-(-3)}{2-6} = \frac{8}{-4} = -2 \]
Subtract the outputs one way and the inputs the other
Why: Both differences are computed correctly in isolation.
The slope is reported as negative 2, so the line is described as falling.
The two subtractions must run in the same direction. Taking the second point first on top and the first point first underneath negates one difference and not the other.
The correct slope is positive 2. A quick sanity check catches it: the second point is both further right and higher than the first, so the line rises and the slope must be positive.
Check the sign against the picture before reporting. Right-and-up means positive, right-and-down means negative, and that check costs nothing.
Prediction
A line passes through a point and then through a second point that is further right and lower.
Predict first
What is the sign of its slope?
Correct: Negative, since the output fell as the input rose.
Why: The run is positive because the second point is further right, and the rise is negative because it is lower. A negative over a positive is negative. Zero slope would need the two points at the same height, and an undefined slope would need them at the same input, which would make the line vertical.
Faded example
Find the slope through the points at (negative 1, 5) and (3, negative 7).
Fill in the blanks
m = \frac-12-3 = \frac___}___ = ___
Why: The outputs drop by 12 while the inputs rise by 4, giving a slope of negative 3. The double negative in the denominator is the usual slip: 3 minus negative 1 is 4, not 2. Checking the sign against the picture confirms it, since the second point is right and well below the first.
Explain it to yourself
Section 1.3 computed the difference quotient for several functions.
Discussion prompt
What did the difference quotient of a linear function come out as, and how does that relate to this section's definition?
Hint: Did it still mention the letter a?
Answer:
It came out as the slope alone, with no a and no h remaining. For the rule with slope 3, the difference quotient simplified to exactly 3.
That is the algebraic statement of this section's definition. The difference quotient IS the average rate of change over the interval from a to a plus h, so its being free of both letters says the rate does not depend on where you measure or how wide the interval is.
Every other function in §1.3 gave a quotient that still mentioned a. Lines were the single exception, and this section is the study of that exception — which is why linear functions are the family everything else gets compared to.
Section
Section 2
Concept
Written as a constant times the input plus a constant, a linear rule displays its slope and its output at zero directly, with no rearranging required.
\[ f(x)=mx+b \]
The phrase 'y-intercept' is worth being careful with. It is a point, at height b on the vertical axis, but it is usually quoted as the single number b. Questions asking for the intercept sometimes want the point and sometimes the number, and reading which is meant saves an unnecessary loss of marks.
Figure (svg): The three standard forms of a linear equation side by side, each with the piece of information it displays directly highlighted
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 190-196
Picture it
Each form gives away a different fact without any work.
Figure (svg): The three standard forms of a linear equation side by side, each with the piece of information it displays directly highlighted
The efficient habit is to pick the form matching what you were given. Given a point and a slope, point-slope is a direct substitution; given the intercept, slope-intercept is.
Worked example
Read the intercept, then step off the slope.
\[ \text{A line crosses the vertical axis at } 3 \text{ and rises } 2 \text{ for every } 1 \text{ across. Write its rule.} \]
Identify the intercept
Why: The output at the input zero.
\[ b = 3 \]
Identify the slope
Why: Rise over run.
\[ m = 2 \]
Substitute into the form
Why: Slope times input, plus intercept.
\[ y = 2 x + 3 \]
Check at the input zero
Why: The rule should return the intercept.
\[ y = 3\text{ at } x = 0 \]
Figure (svg): The solution to Worked example write the rule from a graph shown as a ladder of expressions, one row per legal move
\[ f(x)=2x+3 \]
Verify: test a second point
Why: Stepping one unit right from the intercept should raise the output by 2, giving 5 at the input 1. The rule gives 2 plus 3, which is 5. Two points agreeing is enough to confirm a line, since two points determine one.
\[ 2(1) + 3 = 5 \]
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 191-193
Matching
Rearrange first where necessary.
Match the pairs
Why: The first three are the same line written three ways, which is worth seeing: dividing through or moving a term does not change the line. The fourth looks similar and is a different line, because the x term is on the same side as y with a positive coefficient, so isolating y negates it.
Worked example
The numbers mean something, and saying what is usually part of the question.
\[ \text{A gym charges a } 40 \text{ joining fee plus } 25 \text{ per month. Write and interpret the rule.} \]
Identify what varies
Why: The number of months is the input.
Identify the rate
Why: Cost per additional month.
\[ m = 25\text{ per month} \]
Identify the value at zero months
Why: The fee paid before any months.
\[ b = 40 \]
Assemble the rule
Why: Rate times months, plus fee.
\[ C(t) = 25 t + 40 \]
Figure (svg): The solution to Worked example interpret the two constants in context shown as a ladder of expressions, one row per legal move
\[ C(t)=25t+40 \]
Verify: check both constants mean what you claimed
Why: At zero months the rule gives 40, which is the joining fee alone — correct, since nothing has been used yet. Going from 3 months to 4 raises the cost by 25, which is the monthly rate. Each constant does the job attributed to it, which is the check that the model was assembled correctly rather than just plausibly.
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 194-196
Error analysis
A student reads the slope from an equation not yet in slope-intercept form.
Annotate
On: \( 3x + y = 12 \;\Longrightarrow\; m = 3, \; b = 12 \)
Solve for the output variable before reading any coefficients. A form only displays what it promises when the equation is actually in that form.
Prediction
The slope of a linear rule is kept fixed and its constant is increased by 5.
Predict first
What happens to the graph?
Correct: It shifts up 5.
Why: The constant is added outside the function, so by §1.5 it acts on the output and moves the graph vertically. The slope is untouched, so the steepness does not change and the new line is parallel to the old one. Changing m instead would change the steepness, which is a rotation rather than a shift.
Faded example
Put the equation with 5x plus 2y equal to 8 into slope-intercept form.
Fill in the blanks
2y = -5x + 8 \;\Longrightarrow\; y = -5/2x + 4
Why: Subtracting 5x and then dividing everything by 2 gives a slope of negative five halves and an intercept of 4. Both terms must be divided by the 2, and dividing only the constant is the standard slip. Checking at the input 0 gives 4, confirming the intercept.
Real world
In an applied problem the two constants almost always have names.
Discussion prompt
A phone plan costs a fixed monthly charge plus an amount per gigabyte used. Which constant is which, and what are the units of each?
Hint: Which one is paid even when nothing is used?
Answer:
The fixed charge is b, the value at zero gigabytes, measured in currency. The per-gigabyte amount is m, the slope, measured in currency per gigabyte.
The units are the check. A slope is always output units per input unit, so if the input is gigabytes and the output is dollars, the slope is dollars per gigabyte — which is exactly what a per-unit price is.
This pattern covers most linear models you will meet: b is the starting amount and m is the rate of change. Naming both, with units, is usually worth a mark in itself, and it also catches models assembled with the two constants swapped.
Section
Section 3
Concept
Given a slope and any point on the line, the equation can be written down immediately without solving for the intercept first.
\[ y-y_1 = m(x-x_1) \]
The form is really a rearrangement of the slope formula: the slope between the known point and a general point equals m, and multiplying up gives exactly this. Seeing it that way makes it memorable rather than a fourth thing to recall.
Figure (svg): The three standard forms of a linear equation side by side, each with the piece of information it displays directly highlighted
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 196-200
Picture it
Point-slope form is the slope formula with the denominator cleared.
Figure (svg): Two plotted points joined by a line, with the slope formula written beneath and the rise and run marked on the diagram as the differences of the coordinates
Replacing the second point by a general point and multiplying up turns the formula into an equation that describes every point on the line at once.
Worked example
One substitution, then simplify if the question wants slope-intercept form.
\[ \text{Write the line through } (4,-1) \text{ with slope } 3. \]
Substitute into point-slope form
Why: The point supplies both subscripted values.
\[ y - (-1) = 3(x - 4) \]
Tidy the double negative
Why: Minus a negative is plus.
\[ y + 1 = 3(x - 4) \]
Expand if slope-intercept is wanted
Why: Distribute the 3.
\[ y + 1 = 3 x - 12 \]
Solve for y
Why: Subtract 1.
\[ y = 3 x - 13 \]
Figure (svg): The solution to Worked example from a point and a slope shown as a ladder of expressions, one row per legal move
\[ y+1 = 3(x-4) \quad \text{or} \quad y = 3x-13 \]
Verify: check the given point satisfies it
Why: At the input 4 the rule gives 12 minus 13, which is negative 1 — the point we were given. Any correct answer must pass through the point it was built from, and this is the fastest possible check on the whole calculation.
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 197-198
Faded example
Write the line through the point at (negative 5, 2) with slope 4.
Fill in the blanks
y - 2 = 4(x + 5)
Why: The output coordinate 2 is subtracted on the left. The input coordinate negative 5 is subtracted inside the bracket, and subtracting a negative gives a plus 5. Checking at the input negative 5: the bracket is zero, so y equals 2, which is the given point.
Worked example
Find the slope first, then use either point.
\[ \text{Write the line through } (-2,7) \text{ and } (3,-3). \]
Compute the slope
Why: Outputs over inputs, same order.
\[ m = \frac{-3 - 7}{3 - (-2)} = -2 \]
Pick either point
Why: The first one will do.
\[ \text{use } (-2, 7) \]
Substitute into point-slope
Why: Careful with the double negative.
\[ y - 7 = -2(x + 2) \]
Expand and solve for y
Why: Distribute and collect.
\[ y = -2 x + 3 \]
Figure (svg): The solution to Worked example from two points shown as a ladder of expressions, one row per legal move
\[ y = -2x+3 \]
Verify: check BOTH given points
Why: At negative 2: 4 plus 3 is 7. At 3: negative 6 plus 3 is negative 3. Both points lie on the line, which confirms the slope and the substitution together. Using the other point in the point-slope step gives the same final equation, which is worth trying once to see that the choice genuinely does not matter.
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 199-200
Trap
\[ \text{through } (-2,7) \text{ with } m=-2: \quad y-7 = -2(x-2) \]
Substitute the point into the form
Why: The slope and the output coordinate are placed correctly.
The input coordinate negative 2 is written as a plain 2 inside the bracket.
The form subtracts the coordinate, so a negative coordinate becomes a plus. Substituting negative 2 gives x minus negative 2, which is x plus 2.
The wrong version describes a line through the point at input positive 2, which is a different line entirely.
Check the given point satisfies your answer. The wrong version gives 7 minus 8, which is negative 1 rather than 7 at the input negative 2 — caught in one substitution.
Prediction
A line is being found from two given points, and either may be used in the point-slope step.
Predict first
Do the two choices give different final answers?
Correct: No, they simplify to the same equation.
Why: Both points lie on the same line, and a slope together with any point on a line determines that line uniquely. The intermediate point-slope expressions look different, but expanding and solving for y produces the same slope-intercept equation from either. Trying both once is a good way to convince yourself.
Sorting
Pick the form that already displays what you were given.
Sort into buckets
Sort each situation by the form that needs least work.
Explain it to yourself
Point-slope form is not an independent fact to memorise.
Discussion prompt
Derive point-slope form from the slope formula in two lines.
Hint: Take a general point and the known point, and write the slope between them.
Answer:
Take a general point on the line and the known point. The slope between them must be m, so y minus the known output, over x minus the known input, equals m.
Multiplying both sides by the denominator clears the fraction and gives exactly the point-slope form.
So it is the slope formula with the division undone. That also explains the one restriction: the derivation divides by x minus the known input, so it says nothing about the single point where that is zero — which is harmless, since the form is still satisfied there and vertical lines were already excluded by having no slope at all.
Section
Section 4
Concept
A horizontal line has slope zero and is a perfectly ordinary linear function. A vertical line has no slope at all and is not a function.
The distinction matters more than it looks. A slope of zero is a number and can be used in every formula in this section; an undefined slope cannot be used in any of them, which is why vertical lines need the standard form and why they are excluded whenever a section says 'every non-vertical line'.
Figure (svg): Four lines drawn with different slopes: positive rising, negative falling, zero horizontal, and undefined vertical, each labelled with what its slope is and whether it is a function
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 200-204
Picture it
Positive, negative, zero, and undefined, with what each says about the function.
Figure (svg): Four lines drawn with different slopes: positive rising, negative falling, zero horizontal, and undefined vertical, each labelled with what its slope is and whether it is a function
Only the fourth fails to be a function. The third is often mistaken for it, but a horizontal line passes the vertical line test at every point and is a genuine constant function.
Worked example
One horizontal, one vertical, through the same point.
\[ \text{Write the horizontal and vertical lines through } (5,-2). \]
The horizontal line keeps the output fixed
Why: Every point on it has the same height.
\[ y = -2 \]
Check the point is on it
Why: Its height is negative 2.
The vertical line keeps the input fixed
Why: Every point has the same input.
\[ x = 5 \]
Check again
Why: Its input is 5.
Figure (svg): The solution to Worked example write the two lines through a point shown as a ladder of expressions, one row per legal move
\[ y=-2 \quad \text{and} \quad x=5 \]
Verify: notice which coordinate each uses
Why: The horizontal line's equation uses the point's OUTPUT and the vertical line's uses its INPUT, which is the opposite of what most people first guess. The check is the same either way: substitute the point and see whether the equation holds.
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 201-202
Discrimination
Which of the rise and the run is zero decides it.
Sort into buckets
Sort each line.
Worked example
The formula itself refuses to produce an answer.
\[ \text{Compute the slope through } (3,1) \text{ and } (3,8). \]
Subtract the outputs
Why: There is a genuine rise.
\[ 8 - 1 = 7 \]
Subtract the inputs
Why: Both are the same.
\[ 3 - 3 = 0 \]
Attempt the division
Why: Seven divided by zero.
\[ \frac{7}{0} \]
Conclude
Why: Division by zero has no answer.
Figure (svg): The solution to Worked example why the vertical slope is undefined shown as a ladder of expressions, one row per legal move
\[ \text{undefined: the run is } 0 \]
Verify: contrast with a slope of zero
Why: A horizontal line has the situation reversed: the rise is 0 and the run is nonzero, giving 0 divided by something, which is 0 — a perfectly good number. Zero on the top is fine and zero on the bottom is not, and that asymmetry is the whole difference between the two cases.
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 202-203
Error analysis
A student describes a vertical line.
Annotate
On: \( \text{through } (3,1) \text{ and } (3,8): \quad m = 0, \; \text{so } y = 0x + b \)
Zero slope and undefined slope are opposite situations, and the phrase 'no slope' means the second. A vertical line is also not a function, so no rule of the form output equals something can describe it.
Prediction
A line is drawn perfectly vertically on a set of axes.
Predict first
Is it a function?
Correct: No, it fails the vertical line test.
Why: A vertical line contains many points with the same input and different outputs, so that one input has many outputs — exactly what §1.1 forbids. The second option describes a horizontal line, which is a function whose range is a single value, and the two are easy to swap.
Two truths and a lie
Two of these are true of horizontal and vertical lines and one is false.
Eliminate the wrong options
One of these claims is wrong.
Survives elimination: B
Why: B is the false claim, and it inverts the two cases. A vertical line's run is zero, so its slope is undefined rather than zero; a horizontal line is the one with slope zero, because it is the one whose rise vanishes.
Edge cases
Imagine a line rotating steadily from horizontal towards vertical.
Discussion prompt
What happens to its slope during the rotation, and what does that say about the vertical case?
Hint: Track the run as the line steepens.
Answer:
The slope starts at 0 and grows. As the line approaches vertical, the run for a fixed rise shrinks towards zero, so the slope grows without bound.
At the vertical position the run is exactly zero and the slope is not a large number — it is no number at all. The values were heading off to infinity, and infinity is not a value the slope can take.
This is why 'undefined' is the right word rather than 'infinite'. It also explains why vertical lines have to be excluded from so many statements in this chapter: they are not an awkward extreme case of the formulas, they are outside their reach entirely.
Section
Section 5
Concept
For a linear function the sign of the slope decides increasing or decreasing everywhere at once, and its size decides steepness. There are no intervals to name.
Compare this with §1.3, where increasing and decreasing intervals had to be found and named. For a line, the answer is the whole domain, one way or the other, and the reason is exactly the constancy of the rate. This is what it means for lines to be the simple case.
Figure (svg): Four lines drawn with different slopes: positive rising, negative falling, zero horizontal, and undefined vertical, each labelled with what its slope is and whether it is a function
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 186-190
Picture it
Reading behaviour off a line requires only the sign and size of one number.
Figure (svg): Four lines drawn with different slopes: positive rising, negative falling, zero horizontal, and undefined vertical, each labelled with what its slope is and whether it is a function
None of the first three has a turning point, which is why none of them has a local maximum or minimum. Every question §1.3 asked has a one-word answer here.
Worked example
Sign first, then size.
\[ \text{Compare } f(x)=-4x+1 \text{ and } g(x)=2x+1. \]
Read the signs
Why: One negative, one positive.
Compare the sizes
Why: Four against two, ignoring sign.
\[ | - 4 | > | 2 | \]
Interpret the sizes
Why: The larger magnitude is steeper.
Note the shared constant
Why: Both have the same value at zero.
\[ \text{both meet at } (0, 1) \]
Figure (svg): The solution to Worked example compare two lines shown as a ladder of expressions, one row per legal move
\[ f \text{ decreasing, steeper}; \; g \text{ increasing}; \text{ both through } (0,1) \]
Verify: check the crossing
Why: Both rules give 1 at the input 0, so they genuinely meet there. Since their slopes differ, they cross exactly once — two lines with different slopes always meet at exactly one point, which is the whole content of solving a system in Chapter 9.
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 188-189
Ranking
Steepest first, using the size of the slope.
Put in order
Why: The absolute values are 6, 3, 0.5 and 0, which is the order given. The signs are irrelevant to steepness: the steepest line here falls, and the second steepest rises. A slope of zero is the least steep possible, since a horizontal line has no steepness at all.
Worked example
The sign of the rate is usually the answer the question actually wants.
\[ \text{A tank drains at } 3 \text{ litres per minute from } 200 \text{ litres. Describe the model.} \]
Identify the rate and its sign
Why: Draining means the volume falls.
\[ m = -3 \]
Identify the starting value
Why: The volume before any time passes.
\[ b = 200 \]
Write the rule
Why: Rate times time, plus start.
\[ V(t) = -3 t + 200 \]
Describe the behaviour
Why: A negative slope falls everywhere.
Figure (svg): The solution to Worked example behaviour from a context shown as a ladder of expressions, one row per legal move
\[ V(t)=200-3t, \text{ decreasing everywhere} \]
Verify: check the domain the context allows
Why: The formula gives negative volumes once t exceeds about 66.7 minutes, which is physically impossible — the tank is empty then. So the model's useful domain is from 0 to that value, and the mathematics does not know this. Stating the domain a context implies is part of answering, and §2.3 makes it a habit.
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 189-190
Trap
\[ m_1=-4, \; m_2=2 \;\Longrightarrow\; \text{the second is steeper, since } 2 > -4 \]
Compare the two slopes as numbers
Why: Two is greater than negative four, so the second is taken to be the steeper line.
The line with slope 2 is described as the steeper of the two.
Steepness is about size, not sign. The absolute value of negative 4 is 4, which exceeds 2, so the first line is much steeper.
The sign says which direction it goes, not how sharply. A slope of negative 4 falls steeply; a slope of 2 rises gently.
Compare absolute values for steepness and signs for direction. They are two independent readings of the same number, and mixing them produces answers that are backwards rather than merely imprecise.
Prediction
A linear function has a negative slope.
Predict first
On what intervals is it decreasing?
Correct: On its whole domain.
Why: The rate of change is constant, so it is negative everywhere at once and the function falls throughout. There are no turning points to divide the domain into intervals, and the intercept only decides the line's height, not its direction. This is the contrast with §1.3, where finding the intervals was most of the work.
Matching
One number answers several questions.
Match the pairs
Why: The first two have the same steepness and opposite directions, which is the pairing worth noticing: the sign and the size are independent readings. The third rises, but so gently that over a short window it would look almost horizontal — which is why judging steepness by eye from a plot depends on the axis scales.
Counterexample
A classmate claims a line looks steeper on a graph exactly when its slope is larger.
Discussion prompt
Describe a situation where a line with the smaller slope looks steeper.
Hint: Who chooses the scale on each axis?
Answer:
Plot the rule with slope 0.1 on axes where the vertical unit is a hundred times the horizontal one. It will look nearly vertical, while a line of slope 5 on ordinary axes looks moderate.
Apparent steepness on a page depends on the axis scales, which are a choice made by whoever drew the plot. The slope is a property of the function; the visual steepness is a property of the picture.
This is worth knowing beyond the classroom, since it is one of the standard ways a chart misleads: stretching the vertical axis makes a modest trend look dramatic without altering a single number. Read the axis labels before judging a trend by eye.
Comparison
Fill the blanks from memory. Choosing the right form is most of the efficiency in this section.
Comparison matrix
| slope-intercept | point-slope | standard | |
|---|---|---|---|
| written as | y = mx + b | y - y1 = m(x - x1) | Ax + By = C |
| shows directly | slope and y-intercept | slope and one point | neither, without rearranging |
| best when given | the intercept and the slope | any point and the slope | an equation to tidy |
| handles vertical lines | no | no | yes |
The last row is why the standard form survives at all. It is the least convenient for reading off information and the only one that can write down every line.
Pattern
Whatever you are given, the route is the same.
Step 5's check is the one that catches everything. Any correct line must pass through every point it was built from, and testing that costs one substitution per point.
OpenStax Algebra and Trigonometry 2e, §4.1 Linear Functions §4.1
Check
Same order, top and bottom.
Check your understanding
What is the slope of the line through the points (-1, 4) and (3, -8)?
Answer: A
Why: The outputs drop by 12 and the inputs rise by 4, giving negative 3. The sign is right: the second point is further right and much lower, so the line falls.
Check
Rearrange before reading.
Check your understanding
What is the slope of the line given by 6x + 3y = 9?
Answer: A
Why: Solving for y gives y equals negative 2x plus 3, so the slope is negative 2. Both the 6x and the 9 must be divided by 3 when isolating y.
Check
Which coordinate is fixed?
Check your understanding
What is the equation of the vertical line through the point (4, -7)?
Answer: A
Why: A vertical line has every point at the same input, so its equation fixes x at the point's input coordinate, which is 4. It has no slope and is not a function.
Real world
Linear models are the default first attempt at describing almost any relationship, and knowing when the assumption fails matters as much as knowing how to fit one.
Discussion prompt
A shop notices that sales rose by roughly the same number of units each month for six months and models it with a line. What is the model assuming, and when will it break?
Hint: What does a constant rate of change claim about the future?
Answer:
It assumes the rate of change is constant — the same increase every month, forever. That is exactly the property this section defines, and it is a strong claim about months the shop has not seen.
It will break when something limits growth: market saturation, capacity, a competitor. Then the increases shrink, the rate is no longer constant, and the relationship stops being linear — which is what Chapters 3 and 4 are for.
The practical rule is that linear models are reliable near the data and unreliable far from it. Extrapolating a straight line far beyond the range it was fitted on is the standard way a forecast goes badly wrong, and §2.4 returns to it under the name model breakdown.
Commit first
State your confidence along with your answer.
Predict first
A function's average rate of change is 5 over every interval you test. What follows?
Correct: It is linear with slope 5.
Why: A constant rate of change over every interval is precisely the definition of a linear function, and the constant value is its slope. Nothing further is determined — the intercept is still unknown, so the line could sit anywhere vertically — but the slope and the linearity are both settled. A linear function has no local extrema, since it never turns.
Explain it
The test of understanding is being able to say why, not just what.
Discussion prompt
Explain to a classmate why a constant rate of change forces the graph to be a straight line, using the slope triangles rather than a formula.
Hint: What would a bend in the graph do to the triangles?
Answer:
Draw a slope triangle between any two points on the graph. Its shape — the ratio of rise to run — is the average rate of change over that interval.
If the rate is the same everywhere, then every triangle you can draw has the same ratio, so they are all similar triangles, and the segments joining the points all have the same direction. Segments with the same direction from a common starting point lie along one line.
A bend would mean two stretches with different directions, which would give two triangles with different ratios and therefore two different rates. So a bend and a constant rate cannot coexist, which is the statement the section opens with, seen geometrically.
Exit ticket
One honest answer, so the next lesson can start in the right place.
Predict first
Which idea from this lesson would you most want to see again?
Correct: Any of these is a legitimate answer; the useful one is the honest one.
Why: There is no correct choice here. The third costs marks out of proportion to its difficulty, because the two cases are opposites and easy to swap. The second is where most of the routine work of the next three sections lives, so time spent there is repaid immediately.
Connect it up
One page, drawn from memory, is worth more than rereading the section.
Draw it
Draw one line and mark two slope triangles of different widths on it, writing the rise and run of each and showing the two ratios are equal. Beside it, write the same line in all three forms. Then add two more small sketches: a horizontal line labelled with its slope, and a vertical line labelled with what its slope is and why it is not a function.
If your two triangles have different sizes and the same ratio, you have drawn the definition rather than an example of it.
Recap
Five things, and the first is the definition everything else follows from.
| if you remember one thing | it should be this |
|---|---|
| about the definition | the rate is the same over every interval, which is why it is straight |
| about computing slope | subtract in the same order, top and bottom |
| about the forms | pick the one that already shows what you were given |
| about the special cases | horizontal is slope zero; vertical has no slope and is not a function |
Section 2.2 takes these lines onto the coordinate plane in earnest: graphing them quickly, and deciding when two of them are parallel or perpendicular.
OpenStax, Precalculus, §2.1 Linear Functions §2.1, pp. 182-204 — everything on these slides traces back here
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