SAT Math: Nonlinear functions

Nonlinear functions: quadratics, exponentials, polynomials and transformations.

Advanced Math · Nonlinear functions
Both modules
22 practice questions · 6 with free solutions
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 $a b^{t}$ : $a$ is the starting value; $b > 1$ growth, $0 < b < 1$ decay. A 15% decrease means $b = 0.85$ .
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 $1 + \frac{r}{100}$ , not $\frac{r}{100}$ .
$(x - 5)^{2}$ 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 $f (x) = 2 (x - 1)^{2} - 3$ . What is the minimum value of $f (x)$ ?

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

Answer: A: $- 3$

$2 (x - 1)^{2} \geq 0$ , with equality at $x = 1$ , so the minimum value is $- 3$ .

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 5 are open to everyone; the rest need only a free account. Every answer here was checked independently before it was published.

Question 1

The function $f$ is defined by $f (x) = 5 \cdot 3^{x}$ . If $f (a) = 1215$ , what is the value of $a$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 5

$5 \cdot 3^{a} = 1215$ , so $3^{a} = 243$ and $a = 5$ .

Question 2

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

Student-produced response: type your answer.

Show the answer and solution

Answer: 3

$3 \cdot 3^{a} = 81$ , so $3^{a} = 27$ and $a = 3$ .

Question 3

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?

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

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 4

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?

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

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 5

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?

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

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: $V = 30000 (0.85)^{t}$ .