2.2Graphs of Linear Functions
Graph linear functions using slope-intercept form, intercepts, and tables of values. Read slope and intercepts directly from a graph and write the equation of a line from its graph.
Graphing linear functions and reading equations from graphs are foundational skills for every branch of mathematics. Every data trend, every physics formula, and every economic model starts with a line.
A linear function is a function of the form f(x) = mx + b, where m is the slope and b is the y-intercept. Its graph is always a straight line — the slope controls how steeply the line rises or falls, and the y-intercept tells you where it crosses the vertical axis. In this lesson you will learn three methods for graphing linear functions (slope-intercept, intercepts, and table of values), how to read slope and intercepts directly from a graph, and how to recognize and graph horizontal and vertical lines.
Essential Question
How can you use slope and intercepts to graph any linear function, and how can you reverse the process to write an equation directly from a graph?
Lesson Overview
There are three main methods for graphing a linear function. Method 1 — Slope-Intercept: plot the y-intercept (0, b), then use the slope m = rise/run to find a second point, and draw the line. Method 2 — Intercepts: find the y-intercept by setting x = 0, find the x-intercept by setting y = 0, plot both points, and connect them. Method 3 — Table of Values: choose several x-values, compute the corresponding y-values, plot the points, and draw the line through them. To read a graph: identify where the line crosses the y-axis (y-intercept) and the x-axis (x-intercept), then pick two clear lattice points and compute slope = rise/run. Horizontal lines have the form y = k, slope 0, and are functions. Vertical lines have the form x = h, undefined slope, and are NOT functions (they fail the Vertical Line Test). When graphing from standard form Ax + By = C, the intercept method is usually fastest.
Graphing Checklist
- Identify the form (slope-intercept, standard, etc.)
- Find y-intercept: set x = 0
- Find x-intercept: set y = 0
- Use slope to find additional points
- Draw a straight line through the points
- Label intercepts and at least one other point
Reading a Graph → Equation
- Read the y-intercept b from where line crosses y-axis
- Pick two clear lattice points on the line
- Compute slope m = rise/run (count squares)
- Write y = mx + b
- Verify with a third point on the line
Three Methods for Graphing a Line
Reading Slope and Intercepts from a Graph
y-intercept
Where line crosses y-axis. Set x = 0. Here: (0, −3).
x-intercept
Where line crosses x-axis. Set y = 0. Here: (2, 0).
Slope from graph
Pick two lattice points. Count rise ÷ run. Here: 3/2.
Special Lines: Horizontal and Vertical
Horizontal lines are functions (pass VLT). Vertical lines are NOT functions (fail VLT).
Worked Examples
Graph f(x) = −(2/3)x + 4 using the slope-intercept method. Identify the x-intercept.
y-intercept: b = 4. Plot (0, 4).
Slope m = −2/3: from (0, 4), go right 3 (run) and down 2 (rise) → plot (3, 2).
Go right 3 more, down 2 more → plot (6, 0). Draw the line.
x-intercept: set y = 0: 0 = −(2/3)x + 4 → (2/3)x = 4 → x = 6. x-intercept: (6, 0).
Graph 4x − 2y = 8 using the intercept method.
y-intercept: set x = 0: 4(0) − 2y = 8 → y = −4. Plot (0, −4).
x-intercept: set y = 0: 4x − 2(0) = 8 → x = 2. Plot (2, 0).
Draw a line through (0, −4) and (2, 0).
Check: slope = (0−(−4))/(2−0) = 4/2 = 2. Rewrite: y = 2x − 4 ✓.
Write the equation of the line shown in the graph that passes through (−2, 5) and (4, −1).
Find slope: m = (−1 − 5)/(4 − (−2)) = −6/6 = −1.
Use point-slope with (−2, 5): y − 5 = −1(x − (−2)) = −1(x + 2).
Simplify: y − 5 = −x − 2 → y = −x + 3.
Verify with (4, −1): y = −4 + 3 = −1 ✓.
Graph y = 4 and x = −3 on the same axes. Describe each line.
y = 4: horizontal line. Every point has y-coordinate 4. Slope = 0. Draw a horizontal line at y = 4.
x = −3: vertical line. Every point has x-coordinate −3. Slope = undefined. Draw a vertical line at x = −3.
y = 4 is a function (passes VLT). x = −3 is NOT a function (fails VLT).
The two lines are perpendicular to each other (horizontal ⊥ vertical).
A line has x-intercept (−4, 0) and y-intercept (0, 6). Write its equation and find f(10).
Two points: (−4, 0) and (0, 6).
Slope: m = (6 − 0)/(0 − (−4)) = 6/4 = 3/2.
y-intercept b = 6. Equation: y = (3/2)x + 6.
f(10) = (3/2)(10) + 6 = 15 + 6 = 21.
Guided Practice
Graph f(x) = 3x − 6 using the slope-intercept method. State the x-intercept and y-intercept.
Hint: Plot b = −6 at (0, −6). Slope = 3 = 3/1: go right 1, up 3. For x-intercept, set y = 0 and solve.
Graph 2x + 5y = 10 using the intercept method.
Hint: y-intercept: set x = 0 → 5y = 10 → y = 2. x-intercept: set y = 0 → 2x = 10 → x = 5. Plot (0, 2) and (5, 0).
A line on a graph passes through (0, −2) and (3, 4). Write its equation.
Hint: y-intercept is (0, −2), so b = −2. Find slope: m = (4−(−2))/(3−0) = 6/3 = 2. Write y = mx + b.
Describe the graph of x = 5. Is it a function? What is its slope?
Hint: x = 5 is a vertical line. Every point on it has x = 5. Think about the Vertical Line Test and what slope = rise/run gives when run = 0.
A line has x-intercept (3, 0) and y-intercept (0, −4). Write its equation in slope-intercept form.
Hint: Use the two intercept points to find slope: m = (−4−0)/(0−3). Then use b = −4.
Key Vocabulary
x-Intercept
The point where a line crosses the x-axis. Always has y = 0. Found by setting y = 0 in the equation and solving for x.
y-Intercept
The point where a line crosses the y-axis. Always has x = 0. In y = mx + b, the y-intercept is b.
Intercept Method
Graphing a line by finding and plotting both intercepts. Especially useful for standard form Ax + By = C.
Lattice Point
A point on a graph where both coordinates are integers. Used when reading slope from a graph — count rise and run between two lattice points.
Horizontal Line (y = k)
A line with slope 0. Equation: y = k. Parallel to the x-axis. IS a function.
Vertical Line (x = h)
A line with undefined slope. Equation: x = h. Parallel to the y-axis. NOT a function (fails VLT).
Interactive Practice — 5 Questions
What is the y-intercept of the line 3x − 4y = 12?
What is the x-intercept of the line y = 2x − 8?
A line passes through (0, 5) and (4, 1). What is its equation?
Which statement about x = −7 is TRUE?
A line has x-intercept (6, 0) and y-intercept (0, −2). What is its slope?
Independent Practice
Independent Practice
Graph f(x) = −2x + 5 using the slope-intercept method. Label the x-intercept, y-intercept, and one additional point.
Graph 3x − y = 6 using the intercept method. State both intercepts and the slope.
Write the equation of the line that passes through (0, 7) and (−2, 1).
Graph y = −3 and x = 4 on the same coordinate plane. State the slope of each and whether each is a function.
A line has x-intercept (−5, 0) and y-intercept (0, 2). Write its equation and find f(10).
Common Mistakes
Confusing x-intercept and y-intercept: setting x = 0 to find the x-intercept.
To find the y-intercept, set x = 0. To find the x-intercept, set y = 0. Remember: x-intercept has y = 0, y-intercept has x = 0.
Reading slope from a graph by counting non-integer units or using non-lattice points.
Always pick two clear lattice points (integer coordinates) when reading slope from a graph. Count the exact grid squares for rise and run.
Saying x = 5 has slope 0 (confusing it with y = 5).
y = 5 is horizontal with slope 0. x = 5 is vertical with UNDEFINED slope. Vertical lines have no slope at all — division by zero.
Graphing a negative slope by going left and down instead of right and down.
For slope m = −2/3: go RIGHT 3 (positive run) and DOWN 2 (negative rise). Always move right for positive run to stay consistent.
Math Tips
Intercept method shortcut for Ax + By = C: x-intercept = (C/A, 0), y-intercept = (0, C/B). Just divide C by each coefficient.
When graphing, always plot at least THREE points — two to draw the line and a third to verify. If the third point is not on the line, you made an arithmetic error.
To write an equation from a graph: read the y-intercept directly (where the line crosses the y-axis), then count rise/run between two lattice points for the slope.
A line through the origin (0, 0) has b = 0, so its equation is simply y = mx. The origin is both the x-intercept and y-intercept.
Horizontal lines y = k have domain (−∞, ∞) and range {k}. They are constant functions. Vertical lines x = h are not functions at all.