3.6Writing Linear Functions from Multiple Representations
Whether you start from a table, a graph, two points, or a word problem, the goal is always the same: find the slope and the y-intercept, then write f(x) = mx + b.
Why This Matters
Writing linear functions is a core SAT and ACT skill. You will see tables, graphs, points, and word problems — all asking you to produce an equation. Mastering every representation here means you can answer these questions in any form they appear.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
How Do You Write a Linear Function?
Identify the rate of change and the initial value. Use the rate of change as the slope (m) and the initial value as the y-intercept (b), then write the function as f(x) = mx + b. If the initial value is not given directly, use a known point to solve for b.
Lesson Overview
A linear function has the form f(x) = mx + b. The slope m is the rate of change — how much the output changes per unit of input. The y-intercept b is the initial value — the output when x = 0. You can identify both from a table, a graph, two points, or a verbal description.
Key Vocabulary
Rate of Change (Slope)
How much the output changes for each unit increase in input. In f(x) = mx + b, the slope is m.
Example: In the table x=0,1,2,3; y=4,7,10,13, the rate of change is 3.
Initial Value (Y-Intercept)
The output when the input is 0. In f(x) = mx + b, the y-intercept is b.
Example: If f(0) = 4, then b = 4.
Slope-Intercept Form
The equation of a linear function written as f(x) = mx + b or y = mx + b.
Example: f(x) = 3x + 4
Point-Slope Method
Using a known point (x₁, y₁) and slope m to find b: b = y₁ − m·x₁.
Example: Point (2, 9), slope 3: b = 9 − 3(2) = 3. Equation: f(x) = 3x + 3.
Writing a Linear Function from a Table
Step 1: Calculate Δy/Δx between consecutive rows — it must be constant for the relationship to be linear. Step 2: Read b from the row where x = 0. If x = 0 is not in the table, substitute any known point into y = mx + b and solve for b.
| x | y | Δy | Δx | Δy/Δx |
|---|---|---|---|---|
| 0 | 5 | — | — | — |
| 1 | 8 | +3 | +1 | 3 |
| 2 | 11 | +3 | +1 | 3 |
| 3 | 14 | +3 | +1 | 3 |
Slope: Δy/Δx = 3/1 = 3 (constant ✓)
Y-intercept: when x = 0, y = 5 → b = 5
f(x) = 3x + 5
⚠ If Δy/Δx is not constant, the table is not linear.
Writing a Linear Function from a Graph
Line through (0, 2) with slope 3/2
Step 1: Find the y-intercept — where the line crosses the y-axis. Here: (0, 2) → b = 2.
Step 2: Count rise over run to find slope. From (0,2) to (2,5): rise = 3, run = 2 → m = 3/2.
f(x) = (3/2)x + 2
Tip: Pick two points where the line crosses grid intersections (lattice points) to avoid estimation errors.
Writing a Linear Function from Two Points
When x = 0 is not one of the given points, use the two-step method:
Step 1 — Find slope
m = (y₂ − y₁) / (x₂ − x₁)
Subtract y-values (rise) over x-values (run). Keep the same order.
Step 2 — Solve for b
b = y₁ − m · x₁
Substitute either known point and the slope you found.
Always verify by substituting the second point back into your equation.
Writing a Linear Function from a Verbal Description
In a word problem, look for two key pieces of information: a rate (slope) and a starting value (y-intercept). Words like "per," "each," "every," or "for each unit" signal the slope. Words like "initial," "base," "fee," "starts at," or "when x = 0" signal the y-intercept.
Example: Taxi fare
"A taxi charges $3.50 per mile plus a $2 base fee."
- → Slope: $3.50 per mile → m = 3.50
- → Y-intercept: $2 base fee → b = 2
- C(x) = 3.50x + 2
Representation Organizer
| Representation | How to Find Slope (m) | How to Find b |
|---|---|---|
| Table | Δy ÷ Δx between consecutive rows (must be constant) | Read y when x = 0; or substitute any point into y = mx + b |
| Graph | Count rise ÷ run between two lattice points | Read the y-intercept directly (where line crosses y-axis) |
| Two Points | m = (y₂ − y₁) ÷ (x₂ − x₁) | Substitute one point and m into y = mx + b; solve for b |
| Verbal | Identify the rate ("per unit") → m | Identify the starting value ("base," "initial," "fee") → b |
Worked Examples
A table shows: x = 0, 1, 2, 3; y = 2, 5, 8, 11. Write the linear function.
Find slope: Δy/Δx = (5−2)/(1−0) = 3
Check: (8−5)/(2−1) = 3 ✓ Constant rate of change.
Read y-intercept: when x=0, y=2. So b=2.
Write the function: f(x) = 3x + 2
A line passes through (0, −4) and has slope −2. Write the linear function.
The y-intercept is given directly: b = −4.
Slope is given: m = −2.
Write: f(x) = −2x − 4
Write a linear function through (2, 7) and (5, 13).
Find slope: m = (13−7)/(5−2) = 6/3 = 2
Use point (2, 7): 7 = 2(2) + b → 7 = 4 + b → b = 3
Write: f(x) = 2x + 3
Check with (5,13): f(5) = 2(5)+3 = 13 ✓
Write a linear function through (1, 10) and (4, 4). (Negative slope)
Find slope: m = (4−10)/(4−1) = −6/3 = −2
Use point (1, 10): 10 = −2(1) + b → b = 12
Write: f(x) = −2x + 12
Check: f(4) = −2(4)+12 = 4 ✓
A plumber charges a $45 service fee plus $65 per hour. Write a linear function for total cost C after h hours.
Identify slope: $65 per hour → m = 65
Identify y-intercept: $45 base fee → b = 45
Write: C(h) = 65h + 45
Interpret: slope = hourly rate; y-intercept = service fee
A student claims the table x=0,1,2,3; y=3,7,12,18 can be modeled by f(x) = 4x + 3. Is this correct?
Check rate of change: (7−3)=4, (12−7)=5, (18−12)=6. The differences are NOT constant.
This table is NOT linear — the rate of change is not constant.
f(x) = 4x + 3 is incorrect because the table is not linear.
Math Tips
Always verify the rate of change is constant before writing a linear equation from a table.
If the table does not include x = 0, use any point to solve for b after finding the slope.
A fractional slope like 1/2 means "up 1, right 2" — it is still a valid linear function.
Common Mistakes
Using a y-value as the slope.
Slope is the ratio Δy/Δx — the change in y divided by the change in x.
Finding an inconsistent rate of change and still writing a linear equation.
If Δy/Δx is not constant, the table is not linear. Check all consecutive pairs.
Reversing rise and run: writing m = Δx/Δy.
Slope = rise/run = Δy/Δx. Rise is vertical (y), run is horizontal (x).
Forgetting to solve for b when x = 0 is not in the table.
Substitute any known point into y = mx + b and solve for b.
Confusing the first y-value in a table with the y-intercept.
The y-intercept is the y-value when x = 0. If x = 0 is not in the table, calculate b.
Writing y = mx when the y-intercept is not zero.
Always include b in the equation. y = mx only works when the line passes through the origin.
Guided Practice
Guided Practice Video: Writing Linear Functions from Multiple Representations
Watch the guided practice walkthrough for writing linear functions from multiple representations, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
A table shows: x = 0, 2, 4, 6; y = 1, 7, 13, 19. Write the linear function.
Hint: Find Δy/Δx between consecutive rows. Then read b from the x=0 row.
Write a linear function through (3, 5) and (7, 13).
Hint: Use m = (y₂−y₁)/(x₂−x₁) first. Then substitute one point to find b.
A cell phone plan costs $25 per month plus $0.05 per text message. Write a linear function for monthly cost C after t texts.
Hint: Identify which number is the slope (rate per text) and which is the y-intercept (base monthly cost).
Interactive Practice — 5 Questions
A table shows x=0,1,2,3; y=4,7,10,13. What is the linear function?
What is the slope of the line through (1,3) and (5,11)?
Write a linear function through (2,9) and (6,17). What is the answer?
A plumber charges $50 per hour plus a $30 service fee. Which function models total cost C after h hours?
A table: x=0,2,4,6; y=5,9,13,17. What is the linear function?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
A table: x=0,1,2,3; y=6,9,12,15. Write the linear function.
A table: x=0,1,2,3; y=10,7,4,1. Write the linear function. (Negative slope)
A table: x=0,2,4,6; y=3,8,13,18. Write the linear function.
Write a linear function with slope 4 and y-intercept −3.
Write a linear function through (0, 7) and (3, 1).
Write a linear function through (2, 5) and (6, 13).
Write a linear function through (1, 4) and (4, 13). (Fractional check: verify slope is a whole number)
Write a linear function through (−1, 8) and (3, 0). (Negative slope)
A car rental costs $40 per day plus a $15 insurance fee. Write a linear function for total cost C after d days.
A table: x=1,2,3,4; y=5,9,13,17. Write the linear function. (x=0 not given — solve for b)
Write a linear function through (2, 3) and (5, 3). Describe the slope.
Challenge★ A table: x=0,1,2,3; y=2,5,10,17. A student writes f(x)=3x+2. Is this correct? Explain.
Challenge