Linear Functions

Linear functions are one of the most heavily tested topics in the entire Digital SAT math section. You'll see questions that ask you to build a function from a word problem, pull meaning from an equation's slope or y-intercept, match a table to a graph, evaluate a function at a given input, or work backward from an output to find the input. Expect to encounter 3–5 questions on linear functions across both math modules. Mastering this topic gives you a strong foundation because the skills here (writing equations, interpreting constants, reading graphs and tables) show up repeatedly in different forms.

What Makes a Function "Linear"

A linear function is any function whose graph is a straight line. The general form is:

$f(x) = mx + b$

Here, $x$ is the input, $f(x)$ is the output, $m$ is the slope (also called the rate of change), and $b$ is the y-intercept (the output when the input is 0). The defining feature of a linear function is that the rate of change between any two input output pairs is always the same. If you pick any two rows from a table of values and compute $\frac{\text{change in output}}{\text{change in input}}$, you'll get the same number every time.

You can also see a linear function written as $y = mx + b$, where $y$ plays the role of the output. The SAT uses both notations interchangeably.

Writing a Linear Function from Given Information

Many SAT questions give you information and ask you to write the function rule. There are two common scenarios.

Given Two Input/Output Pairs

If you know two points, you can find the slope first, then solve for $b$.

Example: A linear function passes through $(2, 11)$ and $(5, 23)$. Write the function.

Step 1: Find the slope.

$m = \frac{23 - 11}{5 - 2} = \frac{12}{3} = 4$

Step 2: Plug one point into $f(x) = mx + b$ to find $b$. Using $(2, 11)$:

$11 = 4(2) + b$ $11 = 8 + b$ $b = 3$

The function rule is $f(x) = 4x + 3$.

Given One Point and the Rate of Change

If you already know the slope, you only need one input output pair to find $b$.

Example: A linear function has a rate of change of $-6$ and passes through $(3, 7)$. Write the function.

$7 = -6(3) + b$ $7 = -18 + b$ $b = 25$

The function is $f(x) = -6x + 25$.

Interpreting Linear Functions in Context

The SAT frequently gives you a linear function tied to a real-world scenario and asks what a specific number in the equation means. Getting these right requires connecting each part of the equation to the context.

Slope = Rate of Change per Unit of Input

The slope tells you how much the output changes for each 1-unit increase in the input.

Y-intercept = Starting Value (When Input Is Zero)

The constant $b$ represents the output value when the input is 0.

Example: A tutoring company charges according to the function $C(h) = 40h + 25$, where $C(h)$ is the total cost in dollars and $h$ is the number of hours of tutoring.

• The 40 means the cost increases by $40 for each additional hour of tutoring. That's the hourly rate.
• The 25 means there's a $25 base fee charged even before any tutoring begins (when $h = 0$).

If the SAT asks "What is the best interpretation of the number 40 in this equation?", the correct answer will say something like "the increase in total cost, in dollars, for each additional hour of tutoring." Watch out for answer choices that flip the slope and intercept meanings or describe the right quantity but attach it to the wrong number.

Interpreting Factors and Structure

Sometimes the equation is structured in a way where seeing the form helps. For instance, if a problem states:

$P(t) = 200 + 15(t - 3)$

This tells you that at $t = 3$, the value is 200, and it increases by 15 for each unit increase in $t$. The SAT may ask what the 200 or the 3 represents. The 200 is the output at time $t = 3$, and the 15 is still the rate of change. Recognizing this structure saves you from needing to simplify the equation first.

Evaluating Linear Functions and Finding Inputs

Given an Input, Find the Output

Substitute the input value into the function and compute.

Example: If $f(x) = -3x + 20$, what is $f(4)$?

$f(4) = -3(4) + 20 = -12 + 20 = 8$

Given an Output, Find the Input

Set the function equal to the given output and solve for the variable.

Example: Using the same function, for what value of $x$ does $f(x) = -1$?

$-3x + 20 = -1$ $-3x = -21$ $x = 7$

In Context

Example: A pool is being drained. The amount of water remaining, in gallons, is modeled by $W(t) = 4800 - 120t$, where $t$ is the time in minutes after draining begins.

How much water remains after 15 minutes?

$W(15) = 4800 - 120(15) = 4800 - 1800 = 3000 \text{ gallons}$

After how many minutes will the pool have 1200 gallons remaining?

$4800 - 120t = 1200$

$-120t = -3600$ $t = 30 \text{ minutes}$

Notice how the slope of $-120$ means the pool loses 120 gallons per minute, and the y-intercept of 4800 is the initial amount of water.

Connecting Tables, Graphs, and Algebraic Representations

The SAT tests whether you can move fluidly between different representations of the same linear function. A table, a graph, and an equation can all describe the same relationship.

From a Table to an Equation

Look at how the output changes as the input increases by a consistent amount. That ratio is the slope.

The input increases by 2 each row, and the output increases by 6 each row. So the slope is $\frac{6}{2} = 3$. Since $f(0) = 5$, the y-intercept is 5. The equation is $f(x) = 3x + 5$.

From a Graph to an Equation

Identify two points where the line clearly crosses grid intersections. Calculate the slope, then read the y-intercept (where the line crosses the vertical axis at $x = 0$). If the y-intercept isn't visible on the graph, use the slope and one point to solve for $b$.

Matching Representations (No Context)

Some questions show you a graph and four equations and ask which equation matches. Your strategy: find the slope and y-intercept from the graph, then check the answer choices. A positive slope means the line rises from left to right; a negative slope means it falls. The y-intercept is where the line hits the vertical axis.

Other questions give a table and ask which graph matches. Check whether the slope is positive or negative, then verify one or two specific points from the table against the graphs.

Matching Representations (In Context)

Context-based versions work the same way, but the axes might be labeled with real-world quantities (time, cost, distance, etc.) rather than just $x$ and $y$. The slope still represents the rate of change of the output quantity per unit of the input quantity, and the y-intercept still represents the output when the input is zero.

Example: A table shows that a delivery service charges $8 for 1 mile, $14 for 2 miles, and $20 for 3 miles. Which function models the total charge $C$, in dollars, for $d$ miles?

The rate of change is $\frac{14 - 8}{2 - 1} = 6$ dollars per mile. Using the point $(1, 8)$:

$8 = 6(1) + b$ $b = 2$

The function is $C(d) = 6d + 2$. The 6 represents the charge per mile, and the 2 represents a base fee.

What to Watch For on Test Day

1. Don't confuse slope and y-intercept in interpretation questions. The SAT designs wrong answers that describe the slope's meaning but attach it to the y-intercept's value, and vice versa. Always check which number in the equation you're being asked about.

2. Check the direction of the slope. If a quantity is decreasing over time, the slope must be negative. If an answer choice shows a positive coefficient on the input variable for a decreasing context, eliminate it.

3. When working from tables, verify the rate of change is constant. Confirm that the ratio of output change to input change is the same between every pair of rows. If it's not constant, the relationship isn't linear (and you may be looking at the wrong model).

4. For "find the input" questions, set up the equation carefully. A common trap is to substitute the output value where the input should go. If the question says $f(x) = 50$, you're solving $mx + b = 50$, not computing $f(50)$.

5. Use structure to save time. If an equation is written in an unusual form like $f(t) = 300 + 25(t - 4)$, you can read off that the function equals 300 when $t = 4$ without expanding. The SAT rewards this kind of structural thinking, especially on harder questions.