SAT Math: Nonlinear functions

Nonlinear functions: quadratics, exponentials, polynomials and transformations.

Advanced Math · Nonlinear functions
Both modules
22 practice questions
Calculator allowed (Desmos)

Practise nonlinear functions →Adaptive SAT module

What the test covers

Vertex, intercepts and forms of quadratics
Exponential growth/decay models: reading the base and initial value
Polynomial zeros and factors; function notation and transformations

Key ideas

$a(x-h)^2+k$ has vertex $(h,k)$: the minimum value is $k$ when $a>0$.
Exponential models $ab^t$: $a$ is the starting value; $b>1$ growth, \(0
Factor theorem: $p(a)=0$ exactly when $x-a$ is a factor.
$g(x)=f(x-h)+k$ moves the graph $h$ right and $k$ up.

Common mistakes

Growth of $r\%$ uses $1+\tfrac r{100}$, not $\tfrac r{100}$.
$(x-5)^2$ has its vertex at $x=5$, not $-5$.

Do it in Desmos

Use sliders for unknown constants and click the vertex or intercepts.

IB link: IB SL 2.5-2.11.

Worked example

The function $f$ is defined by $f(x)=2(x-3)^2-8$. What is the minimum value of $f(x)$?

$8$
$-3$
$-8$
$3$

Answer: C: $-8$

$2(x-3)^2\ge0$, with equality at $x=3$, so the minimum value is $-8$.

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

A bacteria culture is modeled by $P(t)=500(1.08)^t$, where $t$ is the time in hours. To the nearest whole number, what is $P(2)$?

$583$
$580$
$1166$
$540$

Show the answer and solution

Answer: A: $583$

$500\times1.08^2=500\times1.1664=583.2\approx583$.

Question 2

A car is worth $24,000 and loses 15% of its value each year. Which function gives its value $V$, in dollars, $t$ years from now?

$V=24000-0.15t$
$V=24000(0.15)^t$
$V=24000(0.85)^t$
$V=24000(1.15)^t$

Show the answer and solution

Answer: C: $V=24000(0.85)^t$

Each year it keeps $100\%-15\%=85\%$ of its value, so it is multiplied by 0.85 each year: $V=24000(0.85)^t$.

Question 3

The graph of $f(x)=x^2+bx+12$, where $b$ is a constant, has its vertex at $x=4$. What is the minimum value of $f(x)$?

Student-produced response: type your answer.

Show the answer and solution

Answer: -4

The vertex is at $x=-\tfrac b2=4$, so $b=-8$. $f(4)=16-32+12=-4$.

Question 4

The polynomial $p(x)=x^3-4x^2+x+6$. Which of the following is a factor of $p(x)$?

$x-2$
$x+3$
$x-1$
$x+2$

Show the answer and solution

Answer: A: $x-2$

By the factor theorem test the roots: $p(2)=8-16+2+6=0$ ✓. $p(-2)=-20$, $p(1)=4$, $p(-3)=-54$. So $x-2$ is a factor.

Question 5

The function $g$ is defined by $g(x)=f(x-2)+3$, where $f(x)=x^2$. What is the vertex of the graph of $y=g(x)$?

$(2, -3)$
$(3, 2)$
$(2, 3)$
$(-2, 3)$

Show the answer and solution

Answer: C: $(2, 3)$

$g(x)=(x-2)^2+3$: the graph of $x^2$ moved 2 right and 3 up, vertex $(2,3)$.

Question 6

The function $f$ is defined by $f(x)=3\cdot2^x$. If $f(a)=96$, what is the value of $a$?

Student-produced response: type your answer.

Show the answer and solution

Answer: 5

$3\cdot2^a=96$, $2^a=32$, $a=5$.

Question 7

The function $f$ is defined by $f(x)=3(x-5)^2+3$. What is the minimum value of $f(x)$?

$-5$
$9$
$5$
$3$

Show the answer and solution

Answer: D: $3$

$3(x-5)^2\ge0$, with equality at $x=5$, so the minimum value is $3$.

Question 8

The function $f$ is defined by $f(x)=2(x-1)^2-3$. What is the minimum value of $f(x)$?

$-3$
$-6$
$-1$
$1$

Show the answer and solution

Answer: A: $-3$

$2(x-1)^2\ge0$, with equality at $x=1$, so the minimum value is $-3$.

Question 9

The function $f$ is defined by $f(x)=3(x-2)^2+6$. What is the minimum value of $f(x)$?

$6$
$18$
$2$
$-2$

Show the answer and solution

Answer: A: $6$

$3(x-2)^2\ge0$, with equality at $x=2$, so the minimum value is $6$.

Question 10

The function $f$ is defined by $f(x)=2(x-5)^2+3$. What is the minimum value of $f(x)$?

$3$
$5$
$6$
$-5$

Show the answer and solution

Answer: A: $3$

$2(x-5)^2\ge0$, with equality at $x=5$, so the minimum value is $3$.

Question 11

The function $f$ is defined by $f(x)=3(x+2)^2+9$. What is the minimum value of $f(x)$?

$9$
$27$
$-2$
$2$

Show the answer and solution

Answer: A: $9$

$3(x+2)^2\ge0$, with equality at $x=-2$, so the minimum value is $9$.

Question 12

A machine is worth $12,000 and loses 10% of its value each year. Which equation gives its value $V$, in dollars, $t$ years from now?

$V=12000-0.1t$
$V=12000(0.9)^t$
$V=12000(0.1)^t$
$V=12000(1.1)^t$

Show the answer and solution

Answer: B: $V=12000(0.9)^t$

Each year it keeps $100\%-10\%=90\%$ of its value, so it is multiplied by 0.9 each year: $V=12000(0.9)^t$.

Question 13

A machine is worth $18,000 and loses 10% of its value each year. Which equation gives its value $V$, in dollars, $t$ years from now?

$V=18000-0.1t$
$V=18000(0.1)^t$
$V=18000(0.9)^t$
$V=18000(1.1)^t$

Show the answer and solution

Answer: C: $V=18000(0.9)^t$

Each year it keeps $100\%-10\%=90\%$ of its value, so it is multiplied by 0.9 each year: $V=18000(0.9)^t$.

Question 14

A machine is worth $40,000 and loses 20% of its value each year. Which equation gives its value $V$, in dollars, $t$ years from now?

$V=40000-0.2t$
$V=40000(0.2)^t$
$V=40000(1.2)^t$
$V=40000(0.8)^t$

Show the answer and solution

Answer: D: $V=40000(0.8)^t$

Each year it keeps $100\%-20\%=80\%$ of its value, so it is multiplied by 0.8 each year: $V=40000(0.8)^t$.

Question 15

A machine is worth $18,000 and loses 25% of its value each year. Which equation gives its value $V$, in dollars, $t$ years from now?

$V=18000(0.75)^t$
$V=18000-0.25t$
$V=18000(0.25)^t$
$V=18000(1.25)^t$

Show the answer and solution

Answer: A: $V=18000(0.75)^t$

Each year it keeps $100\%-25\%=75\%$ of its value, so it is multiplied by 0.75 each year: $V=18000(0.75)^t$.

Question 16

A machine is worth $30,000 and loses 15% of its value each year. Which equation gives its value $V$, in dollars, $t$ years from now?

$V=30000(0.15)^t$
$V=30000(0.85)^t$
$V=30000(1.15)^t$
$V=30000-0.15t$

Show the answer and solution

Answer: B: $V=30000(0.85)^t$

Each year it keeps $100\%-15\%=85\%$ of its value, so it is multiplied by 0.85 each year: $V=30000(0.85)^t$.

Question 17

The function $f$ is defined by $f(x)=5\cdot3^x$. If $f(a)=1215$, what is the value of $a$?

Student-produced response: type your answer.

Show the answer and solution

Answer: 5

$5\cdot3^a=1215$, so $3^a=243$ and $a=5$.

Question 18

The function $f$ is defined by $f(x)=3\cdot2^x$. If $f(a)=48$, what is the value of $a$?

Student-produced response: type your answer.

Show the answer and solution

Answer: 4

$3\cdot2^a=48$, so $2^a=16$ and $a=4$.

Question 19

The function $f$ is defined by $f(x)=2\cdot2^x$. If $f(a)=32$, what is the value of $a$?

Student-produced response: type your answer.

Show the answer and solution

Answer: 4

$2\cdot2^a=32$, so $2^a=16$ and $a=4$.

Question 20

The function $f$ is defined by $f(x)=2\cdot3^x$. If $f(a)=54$, what is the value of $a$?

Student-produced response: type your answer.

Show the answer and solution

Answer: 3

$2\cdot3^a=54$, so $3^a=27$ and $a=3$.

Question 21

The function $f$ is defined by $f(x)=3\cdot3^x$. If $f(a)=81$, what is the value of $a$?

Student-produced response: type your answer.

Show the answer and solution

Answer: 3

$3\cdot3^a=81$, so $3^a=27$ and $a=3$.

Keep going

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