➕ Algebra 1 · Linear Functions
Graphing Linear Equations
Graph a line from a table of values, from slope and y-intercept, or from the intercepts, handle horizontal and vertical lines, and check whether a point is on it.
About standards and checking
Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Three blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.
- F-IF.C.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima.
- A-REI.D.10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
By the end you can…
- Explain that the graph of an equation is the set of all its solutions, and use that idea to test whether a point is on a line.
- Graph a linear equation from a table of values, from slope-intercept form, and from its intercepts.
- Graph horizontal lines y = b and vertical lines x = a, and tell them apart.
Lesson (school style)
The graph of an equation in two variables is a picture of all its solutions: every point (x, y) whose coordinates make the equation true. For a linear equation, that picture is a straight line. You only need two points to draw it, plus a third as a check. This lesson shows three ways to find points (a table of values, the slope and y-intercept, and the intercepts) and then covers two special lines, horizontal and vertical.
What the graph shows
A solution of an equation like y = x + 2 is a pair (x, y) that makes it true. For example, (3, 5) is a solution because 5 = 3 + 2, but (3, 6) is not.
The graph of the equation is every solution plotted on the coordinate plane. For a linear equation, that is a straight line. Two facts follow:
- Every point on the line is a solution of the equation.
- Every solution of the equation is a point on the line.
So to test a point, ask one question: do its coordinates make the equation true? Remember that (x, y) means x first (right or left), then y (up or down).
Method 1: a table of values
Pick several x-values, substitute each one into the equation to find y, and plot the points. Choose easy numbers, and include 0, a positive number and a negative number.
For y = 3 − x:
- x = −1 gives y = 3 − (−1) = 4, the point (−1, 4)
- x = 0 gives y = 3, the point (0, 3)
- x = 2 gives y = 1, the point (2, 1)
Plot the points, line up a ruler with them, and draw the line. Extend it past the points and add arrowheads, because a line goes on forever in both directions.
Method 2: start at the y-intercept, then use the slope
When the equation is in slope-intercept form, y = mx + b, you can read two things straight off it. The y-intercept is b, the y-value where the line crosses the y-axis. That point is (0, b). The slope is m, the rise over the run (see Slope of a Line).
- Plot the point (0, b), where the line crosses the y-axis.
- Write the slope as a fraction, rise/run. From your point, move the run to the right and the rise up (a negative rise means down). Plot the new point.
- Repeat the move once or twice, then draw the line through the points.
A whole-number slope is a fraction over 1: slope 3 is 3/1, so go up 3 and right 1. A negative slope such as −2/5 means down 2 and right 5. You can also go the other way, up 2 and left 5. Both land on the same line.
If the equation is not in this form, rewrite it first. Sometimes you only reorder the terms: y = 4 − x is the same as y = −x + 4. The variable term comes second, but it keeps its sign, so m = −1 and b = 4. If y is not alone, solve for y (see worked example 3).
Method 3: use the intercepts
The y-intercept is the y-value where the line crosses the y-axis, and every point on the y-axis has x = 0. The x-intercept is the x-value where the line crosses the x-axis, and every point on the x-axis has y = 0.
So set x = 0 to find the y-intercept, and set y = 0 to find the x-intercept. For 2x − 3y = 6:
- Set y = 0: 2x = 6, so x = 3. The x-intercept is 3, and the line crosses the x-axis at (3, 0).
- Set x = 0: −3y = 6, so y = −2. The y-intercept is −2, and the line crosses the y-axis at (0, −2).
Plot the points (3, 0) and (0, −2) and draw the line through them. This is quickest when the equation looks like Ax + By = C.
The catch: a line through the origin, like y = 5x (or any Ax + By = 0 with A and B both nonzero), has both intercepts equal to 0, so it crosses both axes at the same point, (0, 0). That gives only one point, so find a second one with a table or the slope.
Horizontal and vertical lines
Some equations name only one variable.
- y = b (like y = 3): every point on the line has y-coordinate 3, whatever x is. The graph is a horizontal line through (0, 3), and its slope is 0.
- x = a (like x = −2): every point on the line has x-coordinate −2, whatever y is. The graph is a vertical line through (−2, 0), and its slope is undefined.
Remember HOY VUX from the slope lesson: Horizontal, zerO slope, Y equals a number. Vertical, Undefined slope, X equals a number.
A horizontal line y = b with b ≠ 0 never meets the x-axis, so it has no x-intercept. A vertical line x = a with a ≠ 0 never meets the y-axis, so it has no y-intercept. A vertical line also cannot be written as y = mx + b, because its slope is undefined.
Is this point on the line?
Substitute the point’s x and y into the equation. If both sides come out equal, the point is on the line. If they do not, it is not.
Put parentheses around negative numbers when you substitute, so the signs stay right.
For y = b or x = a, only the named coordinate matters: (7, 3) is on the line y = 3 because its y-coordinate is 3, whatever its x-coordinate is.
Plot carefully
- Start at the origin. Move along the x-axis first, then up or down for y. The point (4, 1) is right 4, then up 1.
- Choose a scale so all your points fit, and label the axes.
- Plot at least three points. Two points make the line, and the third is a check. If the three do not line up, look for a slip in your arithmetic or your plotting.
- Draw the line with a ruler, extend it with arrowheads, and write its equation next to it.
Key words
- Solution of an equation in two variables
- An ordered pair (x, y) that makes the equation true when you substitute.
- Linear equation
- An equation in two variables whose graph is a straight line, such as y = 2x − 3, 3x + 5y = 15 or y = 4.
- Table of values
- A list of x-values and the matching y-values from the equation. Each row is a point on the graph.
- y-intercept
- The y-value where a line crosses the y-axis. For y = mx + b it is b, and the line crosses at the point (0, b).
- x-intercept
- The x-value where a line crosses the x-axis. If that value is a, the line crosses at the point (a, 0).
- Slope-intercept form
- A way to write a line’s equation as y = mx + b, where m is the slope and b is the y-intercept.
- Ordered pair
- Two numbers written as (x, y). The first says how far right (or left) to go, and the second says how far up (or down).
Worked examples
Example 1. Graph y = 2x − 3 using a table of values.
- Choose easy x-values, including a negative number and 0: −1, 0, 1 and 2.
- Substitute each one. x = −1 gives y = 2(−1) − 3 = −5. x = 0 gives y = −3. x = 1 gives y = −1. x = 2 gives y = 1.
- The points are (−1, −5), (0, −3), (1, −1) and (2, 1).
- Plot them. They line up: each time x goes up by 1, y goes up by 2, which is the slope. Draw a straight line through them with arrowheads at both ends.
Answer: Answer: A straight line through (−1, −5), (0, −3), (1, −1) and (2, 1).
Example 2. Graph y = −(2/3)x + 1 using its y-intercept and slope.
- Read b = 1, so the y-intercept is 1 and the line crosses the y-axis at (0, 1). Plot that point.
- Read the slope: m = −2/3, so the rise is −2 (down 2) and the run is 3 (right 3).
- From (0, 1), go right 3 and down 2 to reach (3, −1).
- For a point on the left, go the other way: left 3 and up 2 from (0, 1) reaches (−3, 3).
- Check (3, −1) in the equation: −(2/3)(3) + 1 = −2 + 1 = −1. It matches. Draw the line through the three points.
Answer: Answer: A line through (−3, 3), (0, 1) and (3, −1). It goes down from left to right.
Example 3. Graph 2x − 4y = 8 by solving for y first.
- The equation is not solved for y, so do that first. Subtract 2x from both sides: −4y = −2x + 8.
- Divide every term on both sides by −4: y = (−2x) ÷ (−4) + 8 ÷ (−4) = (1/2)x − 2.
- Read b = −2 and m = 1/2. The y-intercept is −2, so plot (0, −2).
- The slope 1/2 means up 1 and right 2: from (0, −2) go to (2, −1), then to (4, 0).
- Check (4, 0) in the original equation: 2(4) − 4(0) = 8. It matches.
Answer: Answer: y = (1/2)x − 2, a line through (0, −2), (2, −1) and (4, 0).
Example 4. Graph 3x + 5y = 15 using the intercepts.
- Find the x-intercept by setting y = 0: 3x + 5(0) = 15, so 3x = 15 and x = 5. The x-intercept is 5, and the line crosses the x-axis at (5, 0).
- Find the y-intercept by setting x = 0: 3(0) + 5y = 15, so 5y = 15 and y = 3. The y-intercept is 3, and the line crosses the y-axis at (0, 3).
- Plot both points, (5, 0) and (0, 3), and draw the line through them.
- Check with a third point, (−5, 6): 3(−5) + 5(6) = −15 + 30 = 15. It matches, so it is on the line.
- The line goes down from left to right, which agrees with the slope: down 3 for every 5 to the right, so m = −3/5.
Answer: Answer: A line through (5, 0) and (0, 3).
Example 5. Graph y = −4 and x = 2. Where do the two lines cross?
- For y = −4, every point has y-coordinate −4, so the line passes through (0, −4), (3, −4) and (−5, −4). It is a horizontal line with slope 0.
- For x = 2, every point has x-coordinate 2, so the line passes through (2, 0), (2, 5) and (2, −3). It is a vertical line with undefined slope.
- The lines cross at the one point that satisfies both equations: x = 2 and y = −4.
Answer: Answer: A horizontal line through (0, −4) and a vertical line through (2, 0), crossing at (2, −4).
Example 6. Is (−3, 11) on the line y = −2x + 5? Is (4, −2)?
- Substitute x = −3: y = −2(−3) + 5 = 6 + 5 = 11. The point’s y-coordinate is 11, so both sides match and (−3, 11) is on the line.
- Substitute x = 4: y = −2(4) + 5 = −8 + 5 = −3. The point’s y-coordinate is −2, and −3 ≠ −2, so (4, −2) is not on the line.
- The parentheses around the negative number keep the signs right.
Answer: Answer: (−3, 11) is on the line. (4, −2) is not.
In plain English
A graph is a picture of an equation. Every point on the picture is a pair of numbers that makes the equation true.
For a straight-line equation, you do not need a hundred points. Two points are enough, and a third is a handy check.
You can get points three ways. One: make a table, pick x-values, and work out y. Two: start where the line crosses the y-axis, then use the slope to step up or down and over. Three: find where the line crosses each axis.
Two special cases are easy. The equation y = 5 is a flat line at height 5. The equation x = 5 is an up-and-down line at 5 on the x-axis.
See it
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
A cookie price list
A bakery sells cookies for 2 dollars each and charges 1 dollar for the bag. The cost y of x cookies is y = 2x + 1. The price list is a table of values: 1 cookie costs 3 dollars, 2 cookies cost 5, 3 cookies cost 7, and 4 cookies cost 9.
How it connects: Each row of the price list is a point: (1, 3), (2, 5), (3, 7), (4, 9). They line up because the cost rises by the same 2 dollars for every cookie, which is the slope. The 1-dollar bag is the starting charge: the y-intercept is 1, and the line crosses the y-axis at (0, 1). Limit: you probably buy whole cookies only, so the story uses just some points on the line. The math graph includes every x.
The scout on the map
In a strategy game, a scout starts at map position (0, 2). Each move takes it 3 squares right and 1 square up, so it stands at (3, 3), then (6, 4), then (9, 5).
How it connects: Repeating the same rise-and-run step keeps you on one straight line. Here the slope is 1/3 and the y-intercept is 2 (the scout starts at (0, 2)), so all three positions satisfy y = (1/3)x + 2. That is exactly how you graph from the slope and the y-intercept.
The jogger’s graph
A jogger runs at a steady pace, so her distance (y) rises along a straight line as time (x) passes. Then she stops at a water station. Time keeps going, but her distance stays the same, so that stretch of the graph is perfectly flat.
How it connects: A flat stretch is like y = b: the same y-value for every x, with slope 0 (HOY). A vertical stretch would mean the distance changes while no time passes, which no runner can do. That is the shape of x = a, whose slope is undefined (VUX). Limit: on a motion graph, time always moves left to right. The equation x = a is perfectly good math, but it cannot describe a runner.
The metronome
A metronome set to 80 beats per minute clicks 80 times each minute. After x minutes, the total is y = 80x clicks: 80 after 1 minute, 160 after 2, and 240 after 3.
How it connects: The line passes through the origin (0, 0), because 0 minutes means 0 clicks, and its slope is 80 clicks per minute. Both intercepts are 0, so the line crosses both axes at the same point, (0, 0), and the intercept method gives only one point. Use the slope or a table to get a second one. Limit: a metronome cannot run for negative minutes, so the story uses only the part of the line to the right of the y-axis.
Measure twice, cut once
A carpenter’s proverb says to check your measurements twice before you make a cut you cannot undo.
How it connects: Two points are enough to draw a line, but plot a third before you pick up the ruler. If the three points do not line up, there is a slip in your arithmetic or your plotting, and you catch it before the line is drawn.
X marks the spot
On a treasure map, a mark shows the exact place where the treasure is buried. We say “X marks the spot” when we point out an exact location.
How it connects: On a coordinate grid, a spot needs two numbers, (x, y). An equation is the clue: a spot is on the line only if its two numbers make the clue true, so substitute to test it. One number alone, like x = 4, is not a single spot. It is a whole vertical line of spots.
The stubborn line
Why is the line x = 4 so stubborn? No matter what y says, x refuses to change.
How it connects: In x = 4, every point has x-coordinate 4, while y can be anything. That makes a vertical line through (4, 0) with an undefined slope.
Memory help
Walk, then climb
In (x, y), x comes first: walk along the hallway (right or left) before you climb the stairs (up or down). The point (3, −2) means walk 3 to the right, then go 2 down.
SIT down and pick a method
When you need points for a line, SIT: Slope-intercept (plot (0, b), then rise and run), Intercepts (set one letter to 0), Table (choose x, find y).
The axis rhyme
On the y-axis wall, x is zero, that’s all. On the x-axis floor, y is zero, no more. So for the y-intercept, make x zero. For the x-intercept, make y zero.
Draw the staircase
From your starting point, sketch a small staircase. The sideways step is the run and the up-or-down step is the rise. Write the signs on them before you plot. For slope −3/4, the stair goes right 4 and down 3.
A negative slope goes down as you move right
A slope of −3/4 means down 3 and right 4 (or up 3 and left 4). Going up 3 and right 4 gives the slope +3/4, a different line that tilts the other way. Always read a line from left to right.
The rope and the nails
Pull a rope tight between two nails. A tight rope is straight, so two nails are all it takes to fix it in place. Now hold a third nail against the rope. If the rope touches it, all three nails are lined up. If it misses, one nail is in the wrong place. In the same way, two points fix a line, and a third point checks your work.
Study notes
Key ideas
- The graph of an equation in two variables is the set of all its solutions plotted as points. For a linear equation it is a straight line.
- A point is on the line exactly when its coordinates make the equation true.
- Exactly one straight line passes through two different points, so two points draw the line and a third checks your work.
- Three ways to find points: a table of values, the y-intercept plus the slope, and the two intercepts.
- y = b is a horizontal line (slope 0). x = a is a vertical line (slope undefined).
Steps
- Choose a method that fits the form of the equation.
- Find at least two points, and a third to check.
- Plot the points (x first, then y) and see whether they line up.
- Draw a straight line through them with arrowheads on both ends.
Formulas
- Slope-intercept form: y = mx + b: slope m, y-intercept b (the point (0, b))
- y-intercept: Set x = 0 and solve for y
- x-intercept: Set y = 0 and solve for x
- Horizontal line: y = b, slope 0
- Vertical line: x = a, slope undefined
Watch out
- Plot x first, then y: (3, −2) is right 3 and down 2, not up 3 and left 2.
- A negative slope goes down as you move right (or up as you move left).
- When you substitute a negative number, use parentheses: −2(−3) + 5.
- A line through the origin has both intercepts equal to 0, so it crosses both axes at (0, 0) and you need another point.
- y = 3 is horizontal, not vertical. Think HOY VUX.
Why it works
- Why is the graph of y = mx + b a line? Each time x goes up by 1, y changes by the same amount m. A constant rise over run means a constant steepness, which is what makes a line straight.
- Why do two points draw the line? Exactly one straight line passes through two different points. The third point only checks your arithmetic.
- Why set x = 0 for the y-intercept? Every point on the y-axis has x-coordinate 0.
Common mistakes
Teach it back
Explain to a friend how to graph x + 2y = 8 in two ways (with the intercepts, and with slope-intercept form), and how to check that (2, 3) is on the line.
Explaining it out loud shows what you really know. Did you…
- I found the x-intercept 8 and the y-intercept 4 by setting each variable to 0 in turn, which gives the points (8, 0) and (0, 4).
- I solved for y to get y = −(1/2)x + 4 and used the slope −1/2 (down 1, right 2) from (0, 4).
- I checked (2, 3) by substituting: 2 + 2(3) = 8, so it is on the line.
Your 3-week memory plan
- TodayRead the lesson, then try the flashcards.
- TomorrowRecall it from memory first, then check.
- In 3 daysTake the true/false and practice questions.
- In 1 weekMix it with older topics and teach it back.
- In 3 weeksA last quick review before it fades.
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