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SAT®

Evaluating Linear Functions and Interpreting Them – A Guide

The Albert Team
Last Updated On: June 20, 2025

What We Review

Introduction

Linear functions show up everywhere on the SAT®. Therefore, being able to interpret and evaluate linear functions quickly is a high-value skill. Whether the test gives a story, a table, or a graph, the College Board expects students to switch forms, spot key numbers, and solve in seconds. The following guide breaks the topic into bite-sized pieces, offers step-by-step examples for evaluating linear functions, and finishes with practice questions—perfect for quick review the night before an exam.

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Quick Recap: What Is a Linear Function?

A linear function is a rule that produces a straight-line graph.
It has a constant rate of change called the slope.
Two common equations appear:
- Slope-intercept form $f(x) = mx + b$
- Standard form $Ax + By = C$

Quartl, CC BY-SA 3.0, via Wikimedia Commons

Think of a car’s cruise control. When speed stays at $55$ mph, distance increases by $55$ miles every hour, and the graph of distance vs. time is a straight line.

Building a Linear Model from Context

Creating an equation from words may look long, yet the steps repeat every time.

The Process

Step 1: Identify the variables

Decide which quantity is input ( $x$ ) and which is output ( $f(x)$ ).

Step 2: Find the Rate of Change (Slope)

How much does the output rise or fall when the input increases by one?

Step 3: Locate an Initial Value ( $y$ -Intercept)

This is the output when the input equals zero.

Step 4: Write the Function

Combine slope ( $m$ ) and intercept ( $b$ ) in $f(x) = mx + b$ .

Example: Earning Money at a Part-Time Job

A student earns $\$12$ per hour plus $\$30$ in tips per shift. How much will the student earn in $40$ hours?

Step 1: Variables

Input $x$ = hours worked
Output $f(x)$ = total dollars earned

Step 2: Rate of change

Slope $m = 12$ , representing the dollars per hour

Step 3: Initial value

When $x = 0$ , tips give $\$30$ → $b = 30$

Step 4: Function

f(x) = 12x + 30

Step 5: Determine the pay

f(40)=12(40)+30=480+30=510

She will earn $\$510$ .

Interpreting Pieces of a Linear Function

What the Slope Tells Us

Positive slope → output rises as input rises.
Negative slope → output falls.
Units: “output units per input unit.”

Meaning of the $y$ -Intercept

The situation’s starting point when $x = 0$ .

Interpreting Specific Pairs

Point $(8, 70)$ means “When the input is $8$ , the output is $70$ .” Always include units.

Seeing Structure

Factoring or rearranging may reveal common factors or hidden slopes.

Example 2: Temperature Drop Over Time

The temperature $T$ (in $°C$ ) in a freezer follows $T(h) = -3h + 20$ , where $h$ is time in hours. Determine the temperature after $2$ hours.

Substitute $h=2$ .

T(2)=-3(2)+20=-6+20=14

Thus, the temperature will be $14^\circ C$ .

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Evaluating Linear Functions Efficiently

Substituting

Substitute $x$ into the function.

Mental Math Tricks

If $m$ or $b$ is a multiple of $5$ or $10$ , estimate quickly.
Use distributive shortcuts: $7(50) = 7(5\cdot 10) = 35\cdot 10 = 350$ .

Reverse Evaluation

Sometimes the SAT® gives $f(x)$ and asks for $x$ . Solve the equation for $x$ .

Example: Taxi Fare

A taxi charges according to $C(m) = 2.5m + 4$ , where $m$ is miles and $C(m)$ is cost in dollars.

1. What is the cost for $6$ miles?

Substitute $6$ for $m$ . $C = 2.5(6) + 4 = 19$ .

2. What is the distance for a $\$16$ ride?

Substitute $16$ for $C(m)$ and solve.

Step	Reason
$C(m) = 2.5m + 4$	Original equation
$16 = 2.5m + 4$	Substitute $C(m)=16$
$12 = 2.5m$	Subtract $4$ from both sides
$m = 4.8$	Divide both sides by $2.5$

The distance is $4.8$ miles.

Switching Between Representations

Verbal ↔ Table

Read sentences to fill a table of sample points.

Table ↔ Graph

Plot table points, then connect with a straight line.

Graph ↔ Equation

From a graph, pick two clear points, find the slope, and then the intercept.

Effect of Equation Changes

Increasing $m$ → line gets steeper.
Increasing $b$ → whole line shifts up.

Writing Linear Rules from Limited Info

Two Points

Find slope with $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ .
Insert $m$ and one point into $y - y_1 = m(x - x_1)$ then simplify.

One Point + Slope

Already have $m$ , so substitute directly.

SAT® Shortcut

If one point is the $y$ -intercept, substitute $b$ immediately and save time.

Example: Cell Phone Data Plan

When traveling internationally, a phone plan costs $\$20$ plus $\$8$ per GB. Another plan costs $\$32$ for $2$ GB and $\$48$ for $4$ GB. Write both rules.

For the first plan, there is a flat fee of $\$20$ plus $\$8$ per gigabyte. This gives a slope of $8$ and a $y$ -intercept of $20$ , resulting in the function:

$C_1(g) = 8g + 20$ ,

…where $g$ is the number of gigabytes used and $C_1(g)$ is the total cost for the first plan.

For the second plan, we are given two data points: $(2, 32)$ and $(4, 48)$ .

First, calculate the slope:

$m = \dfrac{48 - 32}{4 - 2} = \dfrac{16}{2} = 8$

Using point-slope form with point $(2, 32)$ :

$y - 32 = 8(x - 2)$

Simplifying: $y = 8x + 16$

So the function rule for the second plan is:

$C_2(d) = 8d + 16$ ,

…where $d$ is the number of gigabytes used and $C_2(d)$ is the total cost for the second plan.

Common Pitfalls & How to Avoid Them

Mixing up a negative slope sign—always double-check units.
Misreading units themselves—hours vs. minutes changes the slope value.
Forgetting consistent variables—do not switch time variable names mid-problem.

Quick Reference Vocabulary Chart

Term	Simple Definition	SAT® Tip
Linear Function	A rule whose graph is a straight line, written in function notation	Look for constant change in tables
Slope ( $m$ )	Change in $y$ per $1$ change in $x$	Include units: “dollars per hour”
$y$ -Intercept ( $b$ )	$y$ -value when $x = 0$	Often hidden in word problems
Input ( $x$ )	Independent variable	Usually time or number bought
Output ( $y$ )	Dependent variable	Cost, distance, temperature, etc.
Rate of Change	Another name for slope	Can be positive or negative
Standard Form	Equation $Ax + By = C$	Good for intercepts

Practice Questions

1. A line passes through $(3, -2)$ and $(7, 10)$ . Write its function in slope-intercept form.

2. For $f(x) = -5x + 40$ , find $x$ when $f(x) = 0$ .

3. A gym charges $\$25$ sign-up plus $\$15$ per month. After how many months will the total cost reach $\$130$ ?

Answers

$y = 3x - 11$
$x = 8$
$7$ months

Final Takeaways & Next Steps

Interpreting and evaluating linear functions means more than crunching numbers. Therefore, remember to:

Spot slope and intercept quickly.
Translate among words, tables, graphs, and equations.
Reverse the process if the SAT® gives outputs first.

Next, time individual practice sets to build speed. Good luck turning every line into easy points!

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