How to solve SAT linear functions questions

A linear function is a rule like $f(x) = mx + b$, where the output changes by the same amount every time the input goes up by 1. On the Digital SAT, linear functions questions ask you to build a function from a table, graph, or description, evaluate it, and explain what the slope and intercept mean in a real situation. Algebra is about 35% of the Math section, or 13 to 15 of the 44 questions, and roughly 2 to 4 of them usually test this skill.

What the questions look like

Interpret the numbers. You get a model such as $C(n) = 2.5n + 40$, where $C$ is the total cost in dollars of printing $n$ shirts. The question asks what $2.5$ or $40$ means. Here $2.5$ is the cost per additional shirt and $40$ is the cost when $n = 0$, a fixed setup fee.

Build the function. You get two points, a table, or a graph and must write the function or use it to find another value. Questions phrased with function notation, like "$f(2) = 11$ and $f(6) = 23$," are the same task.

Use function notation. You are asked for $f(5)$, for the value of $x$ that makes $f(x) = 17$, or for a value of a function defined by a rule like $g(x + 1) = g(x) - 3$.

Step-by-step method

1. Name the input and output and their units. In $h(t)$, the input is $t$ and the output is $h$.
2. Find the slope from two points: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. Output change goes on top.
3. Find the intercept by plugging one point into $y = mx + b$ and solving for $b$. The intercept is the output when the input is 0.
4. Write the function and answer the actual question, which might be a later value, an input, or an interpretation.
5. For interpretation questions, use these templates:
   • Slope: "for each increase of 1 in the input, the output changes by $m$ units."
   • Intercept: "the value of the output when the input is 0."

Worked example

Example 1. A candle burns at a constant rate. Two hours after it is lit, it is 21 centimeters tall. Five hours after it is lit, it is 13.5 centimeters tall. The height $h$, in centimeters, is a linear function of the time $t$, in hours, since the candle was lit.

(a) Write $h(t)$. (b) What does the constant term mean? (c) When will the candle burn out?

The two points are $(2, 21)$ and $(5, 13.5)$.

$$m = \frac{13.5 - 21}{5 - 2} = \frac{-7.5}{3} = -2.5$$

The candle loses 2.5 centimeters per hour. Now find $b$ using $(2, 21)$:

$$21 = -2.5(2) + b \quad \Rightarrow \quad b = 26$$

So $h(t) = 26 - 2.5t$.

The constant term, 26, is the height at $t = 0$. The candle was 26 centimeters tall when it was lit.

The candle burns out when $h(t) = 0$: $26 - 2.5t = 0$, so $t = \frac{26}{2.5} = 10.4$ hours.

Example 2. For the linear function $g$, $g(4) = 10$ and $g(x + 1) = g(x) - 3$ for all $x$. What is $g(0)$?

The rule $g(x + 1) = g(x) - 3$ says that each time the input goes up by 1, the output drops by 3. That is a slope of $-3$. Going from $x = 4$ down to $x = 0$ means moving 4 units left, so the output goes up by $4 \times 3 = 12$:

$$g(0) = 10 + 12 = 22$$

Check by writing the function: $g(x) = -3x + 22$, and $g(4) = -12 + 22 = 10$. It matches.

Common traps

• Flipping the slope. Slope is change in output over change in input. If you divide the time change by the height change in Example 1, you get $-0.4$, which is a common wrong choice.
• Describing the slope as a total. In $C(n) = 2.5n + 40$, the $2.5$ is the cost per shirt, not the total cost of the shirts. Watch for answer choices that say "the total cost" or "the number of shirts" when the coefficient is a rate.
• Confusing $f(3)$ with $f(x) = 3$. $f(3)$ means the input is 3. $f(x) = 3$ means the output is 3 and you must find the input. Read which one the question gives you.
• Missing a scale change. If $t$ is measured in months and the question asks about a year, multiply the monthly slope by 12. If a graph's gridlines count by 5, a rise of 2 boxes is 10 units, not 2.
• Treating the intercept as always meaningful. The intercept is the output at input 0, even if 0 makes no physical sense in the context. Some choices try to describe it as something else, like the starting value at $t = 1$.

Desmos tip

When you have two or more points, Desmos can find the function exactly. Add a table, enter the $x$ values in column $x_1$ and the outputs in $y_1$, then type $y_1 \sim mx_1 + b$ on a new line. Desmos shows the values of $m$ and $b$. For evaluation questions, define the function once, such as $h(t) = 26 - 2.5t$, then type $h(7)$ to get the value directly.