SAT Math: Linear functions

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

Algebra · Linear functions
Both modules
14 practice questions
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 $m=\dfrac{y_2-y_1}{x_2-x_1}$; $f(x)=mx+b$ 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 $b=y_1-mx_1$.

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

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?

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

Answer: D: $65$

The extra 3 hours cost $385-190=195$ dollars, so the rate is $195/3=65$ dollars per hour.

Practice questions

Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Every answer here was re-checked independently by computer before it was published.

Question 1

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

Question 2

The graph of the linear function $y=f(x)$ passes through $(-1,4)$ and $(3,-4)$. What is the $x$-intercept of the graph?

$(-1, 0)$
$(0, 2)$
$(1, 0)$
$(2, 0)$

Show the answer and solution

Answer: C: $(1, 0)$

The slope is $\tfrac{-4-4}{3-(-1)}=-2$, so $y-4=-2(x+1)$, i.e. $y=-2x+2$. $y=0$ at $x=1$: the $x$-intercept is $(1,0)$.

Question 3

A tank holds 1,200 liters of water and drains at a constant 45 liters per minute. Which equation gives the volume $V$, in liters, left in the tank $t$ minutes after it starts to drain?

$V=45-1200t$
$V=1200t-45$
$V=1200+45t$
$V=1200-45t$

Show the answer and solution

Answer: D: $V=1200-45t$

It starts at 1,200 and loses 45 each minute: $V=1200-45t$.

Question 4

For the linear function $f$, $f(6)=24$ and $f(3)=9$. What is the value of $f(0)$?

Student-produced response: type your answer.

Show the answer and solution

Answer: -6

The slope is $\tfrac{9-(24)}{3-6}=5$. Then $f(0)=f(6)-6(5)=-6$.

Question 5

For the linear function $f$, $f(1)=17$ and $f(3)=27$. What is the value of $f(0)$?

Student-produced response: type your answer.

Show the answer and solution

Answer: 12

The slope is $\tfrac{27-(17)}{3-1}=5$. Then $f(0)=f(1)-1(5)=12$.

Question 6

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 $\tfrac{-12-(-8)}{3-1}=-2$. Then $f(0)=f(1)-1(-2)=-6$.

Question 7

For the linear function $f$, $f(1)=6$ and $f(8)=27$. What is the value of $f(0)$?

Student-produced response: type your answer.

Show the answer and solution

Answer: 3

The slope is $\tfrac{27-(6)}{8-1}=3$. Then $f(0)=f(1)-1(3)=3$.

Question 8

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

Student-produced response: type your answer.

Show the answer and solution

Answer: 5

The slope is $\tfrac{3-(-1)}{1-3}=-2$. Then $f(0)=f(3)-3(-2)=5$.

Question 9

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?

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

Show the answer and solution

Answer: B: $65$

The extra 1 hour cost $290-225=65$ dollars, so the rate is $65\div1=65$ dollars per hour.

Question 10

An electrician charges a fixed call-out fee plus an hourly rate. A 2-hour job costs $170 and a 3-hour job costs $225. What is the hourly rate, in dollars?

$85$
$75$
$60$
$55$

Show the answer and solution

Answer: D: $55$

The extra 1 hour cost $225-170=55$ dollars, so the rate is $55\div1=55$ dollars per hour.

Question 11

An electrician charges a fixed call-out fee plus an hourly rate. A 2-hour job costs $140 and a 4-hour job costs $220. What is the hourly rate, in dollars?

$55$
$70$
$60$
$40$

Show the answer and solution

Answer: D: $40$

The extra 2 hours cost $220-140=80$ dollars, so the rate is $80\div2=40$ dollars per hour.

Question 12

An electrician charges a fixed call-out fee plus an hourly rate. A 3-hour job costs $270 and a 5-hour job costs $400. What is the hourly rate, in dollars?

$65$
$90$
$80$
$75$

Show the answer and solution

Answer: A: $65$

The extra 2 hours cost $400-270=130$ dollars, so the rate is $130\div2=65$ dollars per hour.

Question 13

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 $110\div2=55$ dollars per hour.

Keep going

Previous topic: Linear equations in one variable
Next topic: Linear equations in two variables
All SAT Math skills · Adaptive SAT Math module simulator · Official SAT practice (Bluebook, Khan Academy)

SAT is a trademark registered by College Board, which is not affiliated with, and does not endorse, IB Math Revision. Every question here is our own.