SAT Math: Linear functions

Linear functions: slope, intercept and what they mean in context.

Algebra · Linear functions
Both modules
14 practice questions · 4 with free solutions
Calculator allowed (Desmos)

Practise linear functions →Adaptive SAT module

What the test covers

Slope and intercept meanings in context
f(x) = mx + b from a table, graph or two points

Key ideas

Slope $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ ; $f (x) = m x + b$ has $y$ -intercept $b = f (0)$ .
In context, the slope is the rate (per hour, per item) and the intercept is the starting or fixed amount.
From two points: find the slope, then $b = y_{1} - m x_{1}$ .

Common mistakes

Check the units: a fixed fee is the intercept, not the slope.
Subtract the $y$ -values and the $x$ -values in the same order.

Do it in Desmos

Enter the two points as a table and use a regression y1 ~ mx1 + b to read m and b.

Worked example

An electrician charges a fixed call-out fee plus an hourly rate. A 3-hour job costs $225 and a 4-hour job costs $290. What is the hourly rate, in dollars?

$30$
$65$
$\frac{145}{2}$
$75$

Answer: B: $65$

The extra 1 hour cost $290 - 225 = 65$ dollars, so the rate is $65 \div 1 = 65$ dollars per hour.

Practice questions

Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Solutions to the first 3 are open to everyone; the rest need only a free account. Every answer here was checked independently before it was published.

Question 1

An electrician charges a fixed call-out fee plus an hourly rate. A 2-hour job costs $200 and a 4-hour job costs $310. What is the hourly rate, in dollars?

$\frac{155}{2}$
$90$
$100$
$55$

Show the answer and solution

Answer: D: $55$

The extra 2 hours cost $310 - 200 = 110$ dollars, so the rate is $110 \div 2 = 55$ dollars per hour.

Question 2

For the linear function $f$ , $f (2) = 7$ and $f (6) = - 5$ . What is the value of $f (0)$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 13

The slope is $\frac{- 5 - 7}{6 - 2} = - 3$ . Going back 2 units from $x = 2$ : $f (0) = 7 + (- 3) (- 2) = 13$ .

Question 3

For the linear function $f$ , $f (1) = - 8$ and $f (3) = - 12$ . What is the value of $f (0)$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: -6

The slope is $\frac{- 12 - (- 8)}{3 - 1} = - 2$ . Then $f (0) = f (1) - 1 (- 2) = - 6$ .

Question 4

A plumber charges a fixed call-out fee plus an hourly rate. A 2-hour job costs $190 and a 5-hour job costs $385. What is the hourly rate, in dollars?

$60$
$77$
$95$
$65$