linear functions

Linear functions show up everywhere on the SAT—modeling rates, lines, and relationships. This walkthrough gives you a deep mastery of slope, intercepts, function notation, and SAT-style applications with graphs and word problems. Every concept comes with step-by-step breakdowns and practice you can toggle open for insight.

what are linear functions?

A linear function is any function of the form $f(x) = mx + b$ , where $m$ and $b$ are constants. Its graph is always a straight line—no curves, no bends. The slope ( $m$ ) tells you the rate of change, or how fast $y$ increases for every 1-unit increase in $x$ . The y-intercept ( $b$ ) is where the line crosses the $y$ -axis.

For example:

f(x) = 3x - 7

In this function, the slope is 3, so for every 1 unit increase in $x$ , $f(x)$ increases by 3. The y-intercept is –7, meaning the line crosses the $y$ -axis at –7. The graph of $f(x) = 3x - 7$ is a straight line that rises steadily and crosses the $y$ -axis below zero.

forms, slope, and intercept

Linear equations can be written in different ways, and being able to move between these forms is essential for the SAT. One especially common skill is finding the slope when you’re given two points.

Practice Question:

What is the slope of the line that passes through the points $(4, 7)$ and $(-2, -5)$ ?

m = \frac{y_2 - y_1}{x_2 - x_1}

To find the slope, subtract the y-values and x-values in the same order. Here, $y_2 = -5$ , $y_1 = 7$ , $x_2 = -2$ , $x_1 = 4$ .

m = \frac{-5 - 7}{-2 - 4} = \frac{-12}{-6} = 2

The slope is 2. This means for every 1 unit increase in $x$ , the line goes up by 2 units.

Another example:

Convert $3x - 2y = 8$ to slope-intercept form ( $y = mx + b$ ).

3x - 2y = 8 \\ -2y = -3x + 8 \\ y = \frac{3}{2}x - 4

Now you can see the slope is $\frac{3}{2}$ and the y-intercept is –4.

forms, slope, and intercept

Lines can be parallel, perpendicular, horizontal, or vertical depending on their slopes. Parallel lines have the same slope and never intersect. For example, consider $y = 2x + 3$ and $y = 2x - 5$ . Both lines have slope 2, so they are parallel even though they cross the y-axis at different points.

Example (parallel lines):

\text{slope of both lines} = 2

Both lines rise 2 units for every 1 unit increase in x. Their graphs never cross.

Perpendicular lines have slopes that are negative reciprocals. For example, $y = 2x + 1$ and $y = -\frac{1}{2}x + 4$ are perpendicular because 2 and -0.5 multiply to -1:

Example (perpendicular lines):

2 \times \left(-\frac{1}{2}\right) = -1

If the slope of one line is m, the slope of a perpendicular line is -\frac{1}{m}.

A horizontal line has slope 0 and is written as $y = b$ (for example, $y = 4$ ). A vertical line has an undefined slope and is written as $x = a$ (for example, $x = -2$ ).

Example (special cases):

\text{Horizontal: } y = 4 \quad \text{(slope 0)}

\text{Vertical: } x = -2 \quad \text{(undefined slope)}

SAT tip: To find a perpendicular slope, flip the number and switch the sign!

forms, slope, and intercept

Function notation is a way to write and work with functions efficiently. It simply means “plug in for x.” For example, if $f(x) = 2x + 5$ , then to find the value when $x = 4$ , just substitute:

f(4) = 2(4) + 5 = 8 + 5 = 13

Here, the input $x = 4$ gives an output of 13.

If you see something like $f(a+1)$ , just plug in $a+1$ wherever you see $x$ :

f(a+1) = 2(a+1) + 5 = 2a + 2 + 5 = 2a + 7

The result is an expression in terms of $a$ , not a number.

Sometimes SAT problems ask you to “read between the lines”—you may have to infer the slope or y-intercept by looking at a graph or interpreting a word problem, even if the equation isn’t given. And if a line passes through the origin, that means the y-intercept is 0.

word problems

Word problems on the SAT often describe a real situation with numbers and relationships. The key is to translate the words into a math equation, step by step. Here’s a reliable method:

1. Assign a variable. Decide what you’re solving for and let a letter represent it.
2. Write an equation. Model the relationships and actions in the problem using math.
3. Solve and answer. Work through the algebra, then double check you’ve answered what the question actually asks.

Example:

A club charges a $20 membership fee plus $3 for every meeting attended. If Jacob paid $38, how many meetings did he attend?

Let $m$ be the number of meetings.

20 + 3m = 38

The $20 fee plus $3 times the number of meetings equals the total paid.

3m = 38 - 20 = 18

Subtract $20 to isolate the meeting payments.

m = \frac{18}{3} = 6

Jacob attended 6 meetings.

Always check if your answer makes sense in the context of the question!

Let's practice with a few College Board questions!

Question 1

An economist modeled the demand $Q$ for a certain product as a linear function of the selling price $P$ . The demand was 20,000 units when the selling price was $40 per unit, and the demand was 15,000 units when the selling price was $60 per unit. Based on the model, what is the demand, in units, when the selling price is $55 per unit?

Question 2

The cost of renting a backhoe for up to 10 days is $270 for the first day and $135 for each additional day. Which of the following equations gives the cost $y$ , in dollars, of renting the backhoe for $x$ days, where $x$ is a positive integer and $x \\leq 10$ ?

Question 3

If $f(x) = \frac{2x - 1}{3}$ , what is the value of $f(5)$ ?

✓