Chapter 4 of Algebra 1: Concepts and Skills, built for a visual learner. The coordinate plane and its quadrants, graphing from a table, horizontal and vertical lines, intercepts, slope drawn as a staircase, direct variation through the origin, slope-intercept form read straight off the equation, and the vertical line test.
Subject: Algebra 1 · 61 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
Algebra 1 · Chapter 4
The coordinate plane, slope as a staircase, intercepts, and reading a line straight off its equation
Objectives
This is the chapter where algebra becomes something you can see. Every equation from here on has a picture.
Figure (svg): A coordinate plane with the four quadrants labelled one to four and the sign of x and y marked in each
Section
Section 4.1
Concept
A point is named by an ordered pair. The first number is the horizontal position and the second is the vertical one.
Figure (svg): A coordinate plane showing the point three comma two reached by moving three across then two up, with the wrong order shown faded
ordered pair — Two coordinates written in a fixed order, x first and y second. The order is the whole point — swapping them names a different place.
Prediction
Commit before plotting.
Predict first
Are the points (5, 2) and (2, 5) the same point?
Correct: No — they are different points, and they sit on opposite sides of the diagonal.
Why: The point (5, 2) is five across and two up, while (2, 5) is two across and five up. Both are in the first quadrant but they are genuinely different places. The order in an ordered pair is not a convention you could reverse — it carries half the information.
Worked example
Plot the four points below and say which quadrant each lies in.
\[ (3, 2), \; (-4, 1), \; (-2, -3), \; (1, -4) \]
For each point, move across first, then up or down
Why: Reading the pair in order is what keeps the two coordinates from swapping. Negative first coordinate means move left; negative second means move down.
Read the quadrant from the pair of signs
Why: Both positive is quadrant one, and the numbering runs anticlockwise from there.
| point | signs | quadrant |
|---|---|---|
| (3, 2) | plus, plus | I |
| (-4, 1) | minus, plus | II |
| (-2, -3) | minus, minus | III |
| (1, -4) | plus, minus | IV |
Figure (svg): A coordinate plane with four points plotted, one in each quadrant, each labelled with its coordinates
Verify: read each plotted point back off the grid
Why: Counting from the origin to the first point gives three right and two up, which is the pair we were given, so the plotting order was applied correctly.
Error analysis
A student plotted the point below in the wrong place. Diagnose it.
Annotate
On: \( (-3, 5) \;\to\; \text{plotted 5 left and 3 up} \)
A quadrant check confirms the signs. Only recounting confirms the sizes.
Matching
No plotting needed — read the pair of signs.
Match the pairs
Why: The first sign says left or right, the second says down or up. Quadrant numbering starts at the top right and runs anticlockwise, which is why quadrant II is up and to the left rather than down and to the right.
Real world
Ordered pairs are not confined to graph paper.
Discussion prompt
Name two systems outside mathematics that locate something with two ordered numbers, and say what happens if the order is swapped.
Hint: Think about maps, or where a pixel lives on a screen.
Answer:
Map references and latitude/longitude locate a place with two numbers, and swapping them puts you in a completely different part of the world.
Screen pixels in a game or an image are addressed the same way, and a swapped pair draws in the wrong place.
In every case the order is not a formality — it is half the address.
Section
Section 4.2
Concept
A linear equation in two variables has infinitely many solutions, and the line is all of them drawn at once.
Figure (svg): A table of three x and y values beside a coordinate plane where those three points lie on one straight line
Every point on the line makes the equation true. Every point off it makes the equation false. There is no third category.
Worked example
Graph the equation below.
\[ y = 2x + 1 \]
Choose three convenient x values
Why: Pick small numbers including zero and one negative, so the points spread out and mistakes are visible.
Substitute each to find y
Why: Do the arithmetic one row at a time and write the pair down before moving on.
| x | 2x + 1 | y | point |
|---|---|---|---|
| -1 | 2(-1) + 1 | -1 | (-1, -1) |
| 0 | 2(0) + 1 | 1 | (0, 1) |
| 2 | 2(2) + 1 | 5 | (2, 5) |
Plot the three points and draw the line through them
Why: If the three points are not in a straight line, one of them is wrong — that is the whole reason for plotting a third.
Figure (svg): A table of three x and y values beside a coordinate plane where those three points lie on one straight line
Verify: test a fourth point from the line
Why: The line passes through (1, 3), and substituting one into the equation gives two plus one, which is three. The extra point agrees, so the line is drawn correctly.
Explain it to yourself
Two points already determine a line. So why plot three?
Discussion prompt
If two points fix a line completely, what job is the third point doing?
Hint: Could two wrong points ever look wrong?
Answer:
The third point is not defining the line — it is checking it. Any two points you plot will always look like a straight line, even if one of them was computed wrongly, so two points can never expose an arithmetic slip.
Three points can. If they fail to line up, you know immediately that one is wrong, and you know to recheck before drawing anything.
Sorting
A point lies on a line exactly when it makes the equation true. Test each one.
\[ y = 2x + 1 \]
Sort into buckets
Which points lie on this line?
This is the same substitute-and-compare check from Chapter 1, now with two variables instead of one.
Error analysis
One row of this table does not belong to the line. Find it.
Annotate
On: \( y = 3x - 2 \qquad \begin{array}{c|c} x & y \\ \hline 0 & -2 \\ 1 & 1 \\ 2 & 5 \\ 3 & 7 \end{array} \)
A table error is invisible in the arithmetic and obvious in the picture. That asymmetry is why you always plot more points than you need.
Section
Section 4.3
Concept
If an equation pins one variable to a number and never mentions the other, the graph is a straight line parallel to an axis.
Figure (svg): A coordinate plane with a horizontal line at y equals three and a vertical line at x equals minus two
The trick is remembering which is which, and the reliable way is to test a point rather than to recall a rule.
Discrimination
Sort each equation. If you are unsure, ask which letter is stuck.
Sort into buckets
Which way does each line run?
Worked example
Graph both equations on the same axes and name their crossing point.
\[ y = 3 \qquad x = -2 \]
For the first, mark every point at height 3
Why: The equation says nothing about x, so x may be anything. Points such as (0, 3), (5, 3) and (-4, 3) all qualify.
For the second, mark every point 2 to the left
Why: Now y is unrestricted, so (-2, 0), (-2, 6) and (-2, -3) all qualify.
Read off where they meet
Why: The crossing point must satisfy both conditions at once, so its x is negative two and its y is three.
\[ (-2, 3) \]
Figure (svg): A coordinate plane with a horizontal line at y equals three and a vertical line at x equals minus two
Verify: test the crossing point in both equations
Why: The point has y equal to three, satisfying the first, and x equal to negative two, satisfying the second, so it genuinely lies on both lines.
Counterexample
A classmate says: every straight line is the graph of a function.
Discussion prompt
Find a straight line that is not a function, and explain what goes wrong.
Hint: Which line could a vertical ruler touch in more than one place?
Answer:
A vertical line such as x equals two is a counterexample. The input two is paired with every y value at once, so a single input has infinitely many outputs.
This is the origin of the vertical line test: if any vertical line meets a graph more than once, that graph fails the one-output rule. Every other straight line passes.
\[ x = 2 \;\text{ pairs the input } 2 \text{ with every output} \]
Prediction
Commit before drawing.
\[ y = -4 \]
Predict first
What does the graph of this equation look like?
Correct: A horizontal line 4 units below the x-axis.
Why: The equation pins y to negative four and never mentions x, so x may be anything at all. That gives infinitely many points, all at the same height, which is a horizontal line. It is not a single point, because nothing restricts x.
Section
Section 4.4
Concept
The x-intercept is where the line crosses the horizontal axis, and the y-intercept is where it crosses the vertical one.
Figure (svg): A line crossing the x axis at four and the y axis at three, with both intercepts circled
Each is found by setting the other variable to zero, which is the part that gets remembered backwards.
Worked example
Graph the equation below using intercepts only.
\[ 3x + 4y = 12 \]
Set y to zero to find the x-intercept
Why: On the horizontal axis the height is zero, so putting y equal to zero finds where the line crosses it.
\[ 3x = 12 \;\Longrightarrow\; x = 4 \quad \text{so } (4, 0) \]
Set x to zero to find the y-intercept
Why: On the vertical axis the horizontal position is zero, so putting x equal to zero finds that crossing.
\[ 4y = 12 \;\Longrightarrow\; y = 3 \quad \text{so } (0, 3) \]
Plot the two points and join them
Why: Two points fix a straight line exactly, so no table is needed here at all.
Figure (svg): A line crossing the x axis at four and the y axis at three, with both intercepts circled
Verify: test a third point on the drawn line
Why: The line passes through the point where x is 4 over 3 short of the axis; more simply, substituting x equal to 4 and y equal to 0 into the original gives 12, and x equal to 0 with y equal to 3 also gives 12, so both intercepts satisfy the equation.
Trap
Find the x-intercept of 2x plus 5y equals 20.
Set x to zero, because it is the x-intercept
Why: The name has an x in it, so zeroing the x feels like the matching move. This is the single commonest intercept error.
\[ 5y = 20 \;\Longrightarrow\; y = 4 \]
That is the y-intercept, not the x-intercept. The name refers to which axis is crossed, not to which letter is set to zero.
Find the same intercept by thinking about the picture.
On the x-axis, the height is zero, so set y to zero
Why: The x-intercept is a point sitting on the horizontal axis, and everything on that axis has y equal to zero.
\[ 2x = 20 \;\Longrightarrow\; x = 10 \quad \text{so } (10, 0) \]
Say it as a sentence and it stops being confusable: to cross the x-axis, the y must be zero.
Matching
No graphing. Just set each variable to zero in turn.
Match the pairs
Why: In every case the intercept is the constant divided by that variable's coefficient. A larger coefficient pulls its intercept closer to the origin, which is why doubling the coefficient of x halves the x-intercept. The last one has a negative coefficient, so its y-intercept ends up below the origin.
Real world
A phone plan costs 20 dollars a month plus 0.10 dollars per text message.
Discussion prompt
Write the equation, then say in plain English what each intercept means in this story.
Hint: What happens when you send zero texts?
Answer:
\[ C = 0.10t + 20 \]
The y-intercept of 20 is what you pay for sending no texts at all — the fixed monthly charge.
The x-intercept would be the number of texts making the cost zero, which is negative two hundred. That is meaningless here, and its meaninglessness is the point: intercepts are only as real as the situation allows.
Fill the middle
Fill both blanks.
Fill in the blanks
4x - 5y = 20 \;\Longrightarrow\; \text5 (-4, 0) \text___ (0, ___)
Why: Setting y to zero gives 4x equals 20, so x is 5. Setting x to zero gives negative 5y equals 20, so y is negative 4. The negative sign is the part most often lost: dividing 20 by negative 5 must give a negative result.
Section
Section 4.5
Concept
Slope measures steepness: how much the line rises for each unit it runs across.
Figure (svg): A rising line with staircase steps drawn under it showing a rise of two for every run of three
\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \]
Picture it
Before any formula, learn to read the sign of the slope straight off the picture.
Figure (svg): Four small planes showing a positive slope rising, a negative slope falling, a zero slope flat and an undefined slope vertical
The fourth case is genuinely different from the other three: a vertical line has no slope, which is not the same as a slope of zero.
Worked example
Find the slope of the line through the two points below.
\[ (1, 2) \text{ and } (4, 8) \]
Subtract the y values to get the rise
Why: Take them in a consistent order — second point minus first — and keep that order for the run too.
\[ \text{rise} = 8 - 2 = 6 \]
Subtract the x values in the same order to get the run
Why: If you reverse one subtraction and not the other, the sign of the slope comes out backwards.
\[ \text{run} = 4 - 1 = 3 \]
Divide rise by run
Why: The result says the line climbs two units for every one unit across.
\[ m = \frac{6}{3} = 2 \]
Figure (svg): Two points on a plane with the rise of six and run of three drawn as a right triangle under the line
Verify: recompute with the points in the opposite order
Why: Two minus eight is negative six, and one minus four is negative three, and negative six over negative three is still two. The order does not matter as long as it is consistent.
Tweak it
Change m and watch the line pivot. The y-intercept is being held still.
Parameter explorer
Drag the slope. What is the line doing, and what point is it pivoting about?
\[ y = {m}x + 1 \]
Edge cases
Push the slope towards the edge case and watch it break.
Discussion prompt
As a line gets steeper and steeper towards vertical, what happens to the run, and why is a vertical line's slope undefined rather than infinite?
Hint: What is the run between two points that sit directly above each other?
Answer:
The run shrinks towards zero while the rise stays finite, so the fraction rise over run grows without limit.
At exactly vertical the run is precisely zero, and rise divided by zero is not a number at all — there is no value it could sensibly equal. So the slope is undefined, which is a stronger statement than saying it is very large.
\[ m = \frac{\text{rise}}{0} \quad \text{is not a number} \]
Ranking
Steepness is about the size of the slope, not its sign. Order them from flattest to steepest.
Put in order
Why: Steepness is the size of the slope with the sign ignored, so the order is 0, then a half, then 1, then 3, then 5. A slope of negative three is steeper than a slope of positive one, even though it falls rather than rises — the minus sign says direction, not size.
Invariant
Three different staircases are drawn on one straight line. Watch what does not change.
Step through it
One quantity is identical in every frame. Which, and why does that make the line straight?
The ratio is the invariant, and it is what being straight means. A curve is precisely a graph whose rise over run keeps changing.
Section
Section 4.6
Concept
Two quantities vary directly when their ratio never changes. The graph is always a straight line through the origin.
Figure (svg): A line through the origin with the constant of variation labelled, beside a table showing y divided by x is always the same
\[ y = kx \quad \text{where } k \text{ is the constant of variation} \]
Worked example
The quantity y varies directly with x, and y is 15 when x is 3. Find y when x is 8.
Write the direct variation model
Why: Every direct variation has the same shape, so start by writing it before any numbers go in.
\[ y = kx \]
Substitute the known pair to find k
Why: Fifteen equals k times three, so dividing gives the constant.
\[ 15 = 3k \;\Longrightarrow\; k = 5 \]
Use the constant with the new input
Why: The constant is a property of the relationship, so it stays fixed while x changes.
\[ y = 5(8) = 40 \]
Figure (svg): A line through the origin of slope five, with the known point at three comma fifteen and the new point at eight comma forty
Verify: check that both pairs give the same ratio
Why: Fifteen over three is five, and forty over eight is also five. The ratio is unchanged, which is exactly what direct variation claims.
Sorting
Sort each relationship. The test is whether the graph passes through the origin with a constant ratio.
Sort into buckets
Which of these vary directly?
Explain it to yourself
Direct variation graphs always go through (0, 0). Say why that is forced rather than chosen.
Discussion prompt
Using the equation y equals k x, explain why the graph must pass through the origin.
Hint: Put x equal to zero into the equation and see what y is forced to be.
Answer:
Substitute zero for x: k times zero is zero, whatever k is. So the pair (0, 0) always satisfies the equation, and the graph always contains that point.
This is also the practical test. If a relationship has any fixed amount that applies at zero input — a delivery fee, a pickup charge, a starting height — then it is not direct variation, because the graph misses the origin.
Tweak it
Every one of these lines goes through the origin. Only the steepness changes.
Parameter explorer
Drag k. What changes, and what refuses to change?
\[ y = {k}x \]
Section
Section 4.7
Concept
When an equation is written as y equals m x plus b, the two numbers are the slope and the y-intercept, ready to read.
Figure (svg): The equation y equals m x plus b with arrows labelling m as the steepness and b as where the line crosses the y axis
No table, no substitution. Start at b on the vertical axis, then step out using the slope.
Worked example
Graph the equation below without making a table.
\[ y = \frac{2}{3}x - 1 \]
Plot the y-intercept first
Why: The constant is negative one, so the line crosses the vertical axis at the point (0, -1). That is your starting dot.
Use the slope as a staircase from that dot
Why: The slope is two over three, so from the starting dot move three to the right and two up, and mark a second dot.
Repeat once more, then draw the line
Why: A third dot from a second staircase step confirms the first two are right before you commit to a line.
Figure (svg): A line drawn from its y-intercept using two staircase steps of run three and rise two
Verify: check one staircase point in the equation
Why: The point (3, 1) should satisfy the equation: two thirds of three is two, and two minus one is one, which matches. The staircase landed on the real line.
Fill the middle
Rearrange into slope-intercept form and read the pair.
Fill in the blanks
2x + y = 7 \;\Longrightarrow\; y = -2x + 7 \;\text-2 m = 7 \text___ b = ___
Why: Subtracting 2x from both sides isolates y and gives negative two x plus seven. The slope is negative two, so the line falls two units for every one across, and it crosses the vertical axis at seven. Reading the slope as positive two is the usual slip — the sign belongs to the coefficient.
Comparison
Fill the blanks. Each method is fastest in a different situation.
Comparison matrix
| method | what you need first | best when |
|---|---|---|
| table of values | three chosen x values | the equation is in an awkward form |
| intercepts | set each variable to zero in turn | the equation looks like ax + by = c |
| slope-intercept | m and b read straight off | the equation is already y = mx + b |
Pattern
One procedure that covers every case in this chapter.
Discrimination
Four equations, four descriptions. Match by reading m and b, without drawing anything.
Sort into buckets
Sort each equation by the picture it describes.
Two truths and a lie
Three claims about slope-intercept form.
Eliminate the wrong options
Which statement is false?
Survives elimination: C
Why: Keep the false statement, which is C. A vertical line such as x equals four cannot be written that way at all, because it has no slope and its equation never mentions y. Every other straight line does fit the form, which is why the exception is so easy to forget.
Section
Section 4.8
Concept
A relation is any set of ordered pairs. A function is a relation where each input appears only once.
Figure (svg): A large oval labelled relations containing a smaller oval labelled functions, showing functions are a special kind of relation
vertical line test — If any vertical line crosses a graph more than once, the graph is not a function — that crossing is one input with two outputs.
Worked example
Decide whether each graph below is a function.
Imagine sliding a vertical ruler across the graph
Why: The ruler represents choosing one input and asking how many outputs it has.
Count crossings at the worst place, not the easiest
Why: A graph only needs to fail once to fail entirely, so look for the place where the ruler hits most.
Figure (svg): Three graphs tested with a dashed vertical ruler: a line passes, a sideways parabola fails with two crossings, and a vertical line fails badly
Verify: name the offending input on each failing graph
Why: On the sideways curve, the input zero produces both a positive and a negative output. On the vertical line, one input produces every output at once. Both have a genuine input with two answers.
Definition probe
Sort each set of pairs by whether it qualifies as a function.
Sort into buckets
Which of these relations are functions?
Explain it
A classmate has memorised the vertical line test but cannot say why it works.
Discussion prompt
Explain what a vertical line represents on a graph, and why more than one crossing means the graph is not a function.
Hint: What do all the points on a vertical line have in common?
Answer:
A vertical line is the set of all points with one fixed x value — in other words, it asks the question: what outputs does this single input have?
Each crossing is one output for that input. Two crossings therefore means one input with two different outputs, which is precisely the thing the definition of a function forbids. The test is not a trick; it is the definition drawn on paper.
Check
Paper first, then click.
Check your understanding
What is the slope of the line through (2, 7) and (5, 1)?
Answer: A
Why: The rise is 1 minus 7, which is negative 6, and the run is 5 minus 2, which is 3. Dividing gives negative 2, so the line falls two units for every one across. A negative slope is expected here because the y value dropped as x increased.
Check
One more, on reading a line off its equation.
Check your understanding
The line 4x - 2y = 8 has which slope and y-intercept?
Answer: A
Why: Solving for y gives negative 2y equals negative 4x plus 8, and dividing by negative 2 gives y equals 2x minus 4. So the slope is 2 and the y-intercept is negative 4.
Commit first
Answer, then rate your confidence.
\[ \text{The line } y = -3x + 5 \]
Predict first
Starting from the y-intercept, which staircase move lands you on another point of this line?
Correct: Right 1, down 3.
\[ m = -3 = \frac{-3}{1} \quad \text{run } 1, \text{ rise } -3 \]
Why: A slope of negative three means the rise is negative three for a run of one, so moving one across sends you three down. Writing the slope as a fraction over one makes this visible: negative three is negative three over one. The commonest error is treating the three as the run.
Exit ticket
Last commitment of the chapter.
Predict first
Which of these is shakiest for you right now?
Correct: Whatever you picked is the one to practise first.
Why: Chapter 5 assumes all four of these are automatic, because it spends its whole length writing equations from slopes and points. A gap here becomes a much bigger gap one chapter later, so it is worth ten minutes now rather than an hour in three weeks.
Connect it up
One page linking every idea to the picture it describes.
Draw it
Draw a coordinate plane in the centre. Around it, attach: ordered pair, table of values, horizontal line, vertical line, x-intercept, y-intercept, slope, direct variation, slope-intercept form, vertical line test. On each arrow write what that idea tells you about the picture.
If slope is not connected to at least three other ideas on your map, look again — it is the hub of this chapter.
Recap
Every equation in this course now has a shape you can draw, and every shape has an equation you can read.
| if you remember one thing | it should be |
|---|---|
| about points | across first, then up — the order carries half the meaning |
| about intercepts | to cross an axis, the other coordinate must be zero |
| about slope | rise over run, and keep both subtractions in the same order |
| about lines | b says where it starts, m says how fast it climbs |
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