Digital SAT Math Practice Questions - Advanced : Nonlinear functions

SAT MAth Practice questions – all topics

Advanced Math Weightage: 35% Questions: 13-15
- Equivalent expressions
- Nonlinear equations in one variable and systems of equations in two variables
- Nonlinear functions

SAT MAth and English – full syllabus practice tests

[calc] Question Hard

The function $f$ is defined by $f(r)=(r-1)(r+2)^2$. If $f(h-5)=0$, where $h$ is a constant, what is one possible value of $h$ ?

▶️Answer/Explanation

Ans: 3 , 6

### Question 19:
To find one possible value of $h$ given $f(h-5) = 0$, we first substitute $h-5$ for $r$ in the function $f(r)$ and set it equal to 0:

\[
f(r) = (r-1)(r+2)^2
\]
\[
f(h-5) = ((h-5)-1)((h-5)+2)^2 = 0
\]
\[
((h-6)(h-3)^2 = 0
\]

Now, we have a product equal to zero, which means at least one of the factors must equal zero. So, we set each factor equal to zero and solve for $h$:

1. $(h-6) = 0$:
\[
h – 6 = 0
\]
\[
h = 6
\]

2. $(h-3)^2 = 0$:
\[
h – 3 = 0
\]
\[
h = 3
\]

So, possible value of $h$ is $h = \boxed{6, 3}$.

[Calc] Question Hard

\[
(x-5)^2+1=0
\]

How many distinct real solutions does the given equation have?
A) Zero
B) Exactly one
C) Exactly two
D) Infinitely many

▶️Answer/Explanation

Ans:A

To determine how many distinct real solutions the equation $(x-5)^2 + 1 = 0$ has,

\[
(x-5)^2 + 1 = 0
\]

First, isolate $(x-5)^2$ by subtracting 1 from both sides:
\[
(x-5)^2 = -1
\]

Now, observe the equation $(x-5)^2 = -1$. Since $(x-5)^2$ represents a square of a real number, it is always non-negative (i.e., it can never be less than zero). The smallest value $(x-5)^2$ can achieve is 0, and it cannot be negative.

However, the right side of the equation is -1, which is a negative number. Since a square of any real number cannot be negative, there are no real values of $x$ that satisfy the equation $(x-5)^2 = -1$.

Therefore, the equation has no real solutions.

The correct answer is:A) Zero

[calc] Question Hard

The function $f$ is defined by $f(r)=(r-1)(r+2)^2$. If $f(h-5)=0$, where $h$ is a constant, what is one possible value of $h$ ?

▶️Answer/Explanation

Ans: 3 , 6

### Question 19:
To find one possible value of $h$ given $f(h-5) = 0$, we first substitute $h-5$ for $r$ in the function $f(r)$ and set it equal to 0:

\[
f(r) = (r-1)(r+2)^2
\]
\[
f(h-5) = ((h-5)-1)((h-5)+2)^2 = 0
\]
\[
((h-6)(h-3)^2 = 0
\]

Now, we have a product equal to zero, which means at least one of the factors must equal zero. So, we set each factor equal to zero and solve for $h$:

1. $(h-6) = 0$:
\[
h – 6 = 0
\]
\[
h = 6
\]

2. $(h-3)^2 = 0$:
\[
h – 3 = 0
\]
\[
h = 3
\]

So, possible value of $h$ is $h = \boxed{6, 3}$.

[calc] Question Hard

\[
f(x)=\left(\frac{3}{4}\right)^x+5
\]

If the function $f$ is graphed in the $x y$-plane, where $y=f(x)$, what is the $y$-intercept of the graph?
A) $\left(\frac{3}{4}, 5\right)$
B) $\left(\frac{3}{4}, 0\right)$
C) $(0,5)$
D) $(0,6)$

▶️Answer/Explanation

Ans: D

The function given is:
\[
f(x) = \left( \frac{3}{4} \right)^x + 5
\]

To find the $y$-intercept, we evaluate $f(x)$ at $x = 0$:
\[
f(0) = \left( \frac{3}{4} \right)^0 + 5
\]
\[
f(0) = 1 + 5
\]
\[
f(0) = 6
\]

So, the $y$-intercept is $(0, 6)$.

Thus, the correct answer is:
\[
\boxed{(0, 6)}
\]

[Calc] Question Hard

Two numbers, $a$ and $b$, are each greater than zero, and the square root of $a$ is equal to the cube root of $b$. For what value of $x$ is $a^{2 x-1}$ equal to $b$ ?

▶️Answer/Explanation

Ans:1.25 or 5/4

\[
\sqrt{a}=\sqrt[3]{b}
\]

We want to find the value of $x$ such that $a^{2 x-1}=b$.

Using the given relationship, we can rewrite $a$ and $b$ in terms of each other:
\[
\sqrt{a}=\sqrt[3]{b} \Longrightarrow a=(\sqrt[3]{b})^2
\]

Now, substitute $a$ into the equation $a^{2 x-1}=b$ :
\[
\left((\sqrt[3]{b})^2\right)^{2 x-1}=b
\]

$ \begin{aligned} & b^{\frac{4 x-2}{3}}=b \\ & 4 x-2=1 \\ & x=3 / 4\end{aligned} $

[Calc] Question Hard

A psychologist conducting a memory experiment provided participants with a list of three-letter sequences. Immediately after the experiment, the participants remembered $100 \%$ of the sequences. The psychologist found that the percentage of sequences the participants remembered decreased by $30 \%$ for every 3 -second interval that passed. Which function best models this situation, where $\mathrm{P}$ is the percentage of sequences the participants remembered, and $t$ is the time, in seconds, that passed?
A) $P(t)=100(0.30)^{3 t}$
B) $P(t)=100(0.30)^t$
C) $P(t)=100(0.70)^{\frac{t}{3}}$
D) $\quad P(\mathrm{t})=100(0.70)^t$

▶️Answer/Explanation

Ans:C

The psychologist’s experiment describes an exponential decay, where the percentage of sequences remembered decreases by 30% every 3 seconds. This means the percentage remaining after each 3-second interval is 70% (or 0.70).

The correct function is:
\[
P(t) = 100(0.70)^{\frac{t}{3}}
\]

[Calc] Question Hard

The population, in millions, of Suzhou, China, can be modeled by the function $p(t)=1.1(1.066)^t$, where $t$ represents the number of years after 1990, and $0 \leq t \leq 25$. Which of the following equations best models the population, in millions, of Suzhou, where $n$ represents the number of years after 1995 , and $0 \leq n \leq 25$ ?
A) $r(n)=1.1(1.066)^{5 n}$
B) $r(n)=1.1(1.066)^{n-5}$
C) $r(n)=1.1(1.066)^5(1.066)^n$
D) $r(n)=(1.1)^5(1.066)^5(1.066)^n$

▶️Answer/Explanation

To find the correct equation that models the population of Suzhou, China, in terms of $ n $, where $ n $ represents the number of years after 1995, we need to adjust the original equation $ p(t) = 1.1(1.066)^t $.

Given:
$ t $ is the number of years after 1990.
$ n $ is the number of years after 1995.

We can express $ t $ in terms of $ n $:
\[ t = n + 5 \]

substitute $ t = n + 5 $
\[ p(t) = 1.1(1.066)^t \]
\[ p(n + 5) = 1.1(1.066)^{n + 5} \]

simplify this equation:
\[ p(n + 5) = 1.1(1.066)^{n + 5} \]
\[ p(n + 5) = 1.1(1.066)^5(1.066)^n \]

Therefore, the equation that models the population in terms of $ n $ is:
\[ r(n) = 1.1(1.066)^5(1.066)^n \]

So the correct answer is:
\[ \boxed{C) \ r(n) = 1.1(1.066)^5(1.066)^n} \]

[Calc] Question Hard

For the quadratic function $h$, the table gives three values of $x$ and their corresponding values of $h(x)$. At what value of $x$ does $h$ reach its minimum?
A. -1
B. 0
C. 3
D. 4

▶️Answer/Explanation

Ans:C

To determine at what value of $x$ the quadratic function $h(x)$ reaches its minimum, we need to look for the vertex of the parabola defined by the table.

Given the values of $h(x)$ for $x = 2$, $x = 4$, and $x = 6$, we can notice that the function $h(x)$ has a symmetric shape, typical of a quadratic function. Since the values at $x = 2$ and $x = 4$ are both $0$, and the value at $x = 6$ is $8$, we can infer that the vertex of the parabola lies between $x = 2$ and $x = 6$.

The vertex of a quadratic function in the form $h(x) = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. However, since we don’t have the exact equation of $h(x)$, we can still find the $x$-coordinate of the vertex by taking the average of the $x$-values where $h(x)$ is $0$. In this case, that would be the average of $2$ and $4$, which is $3$.

Therefore, the minimum value of $h(x)$ occurs at $x = 3$. So, the correct answer is option C: $3$.

[Calc] Question Hard

p(x) = (x + l)(x + 2)(x + 3)
If the given function p is graphed in the xy – plane, Where y = p(x), what is an x-intercept of the graph ?
A) (-6, 0)
B) (-3, 0)
C) (3, 0)
D) (6, 0)

▶️Answer/Explanation

B) (-3, 0)

Given the function $p(x) = (x+1)(x+2)(x+3)$, we need to find an $x$-intercept of the graph where $y = p(x)$.

1. Determine the $x$-intercepts by setting $p(x) = 0$:
\[
(x+1)(x+2)(x+3) = 0
\]

2. Solve for $x$:
\[
x+1 = 0 \quad \Rightarrow \quad x = -1
\]
\[
x+2 = 0 \quad \Rightarrow \quad x = -2
\]
\[
x+3 = 0 \quad \Rightarrow \quad x = -3
\]

Thus, the $x$-intercepts are $(-1, 0)$, $(-2, 0)$, and $(-3, 0)$.

[No calc] Question Hard

The cost of renting a bicycle is $8 for the first hour plus $4 for each additional hour. Which of the following functions gives the cost 𝐶(ℎ), in dollars, of renting the bicycle for h hours, where h is a positive integer?
A. 𝐶(ℎ) = 8ℎ − 4
B. 𝐶(ℎ) = 8ℎ + 12
C. 𝐶(ℎ) = 4ℎ + 8
D. 𝐶(ℎ) = 4ℎ + 4

▶️Answer/Explanation

Ans: D

The cost of renting a bicycle consists of a fixed cost of $\$ 8$ for the first hour and an additional cost of $\$ 4$ for each additional hour. Therefore, the cost function $C(h)$ for renting the bicycle for $h$ hours can be represented as:
\[
C(h)= \begin{cases}8 & \text { if } h=1 \\ 8+4(h-1) & \text { if } h>1\end{cases}
\]

This simplifies to $C(h)=4 h+4$, option D.

[No calc] Question Hard

In the xy-plane, the graph of $y=(-14)(\frac{1}{2})^{x}+k$, where k is a constant, has a y-intercept of (0,2). What is the value of k?

▶️Answer/Explanation

Ans: 16

We’re given that the graph of the equation $ y = (-14) \left( \frac{1}{2} \right)^x + k $ has a $ y $-intercept at $(0, 2)$. This means that when $ x = 0 $, $ y = 2 $.

Substituting $ x = 0 $ and $ y = 2 $ into the equation, we can solve for $ k $:

\[ 2 = (-14) \left( \frac{1}{2} \right)^0 + k \]
\[ 2 = -14 \cdot 1 + k \]
\[ 2 = -14 + k \]
\[ k = 2 + 14 \]
\[ k = 16 \]

So, the value of $ k $ is $ 16 $.

[Calc] Question Hard

A cottonwood tree had a trunk diameter of 2.00 inches at the time it was planted. The tree had a trunk diameter of 22.24 inches x years after it was planted. The equation 0.64𝑥𝑥 + 2 = 22.24 represents this situation. Which statement is the best interpretation of 0.64𝑥𝑥 in this context?

A. The total increase of the tree’s trunk diameterx years after it was planted
B. The tree’s trunk diameter x years after it was planted
C. The maximum trunk diameter of the tree over its lifetime
D. The total increase of the tree’s trunk diameter each year after it was planted

▶️Answer/Explanation

Ans: A

The equation given, $0.64x + 2 = 22.24$, represents the situation where $x$ years after the tree was planted, its trunk diameter increased from its initial diameter by $0.64x$ inches, and the total diameter including the initial diameter is $22.24$ inches.

Therefore, the best interpretation of $0.64x$ in this context is:

A. The total increase of the tree’s trunk diameter $x$ years after it was planted.

So, the correct answer is option A.

[Calc] Question Hard

The graph shown models the height $y$, in feet, of a volleyball $x$ seconds after it was hit by a player. Which equation represents the relationship between the height of the volleyball and the time since the volleyball was hit?
A) $y=-16 x^2+5$
B) $y=-16(x-5)^2$
C) $y=-16(x-0.86)^2$
D) $y=-16(x-0.25)^2+6$

▶️Answer/Explanation

[Calc] Question Hard

The function $f$ is defined by $f(x)=(-8)(6)^x-4$. What is the $y$-intercept of the graph of $y=f(x)$ in the $x y$ plane?
A) $(0,-12)$
B) $(0,-8)$
C) $(0,-4)$
D) $(0,6)$

▶️Answer/Explanation

Question

A scientist tested a group of adults aged 30 to 85. The graph shows the quadratic function $S$, which models their scores on a language test as a function of their age $x$, in years. Which of the following could define $S$ ?

$S(x)=-\frac{1}{320}(x-50)^2+55$
$S(x)=-\frac{1}{320}(x-55)^2+50$
$S(x)=\frac{1}{320}(x-50)^2+55$
$S(x)=\frac{1}{320}(x-55)^2+50$
▶️Answer/Explanation
Ans: B

Questions

The graph of the cubic function $f$ is shown in the $xy$-plane above. If $f(a)=0$, where $a$ is a constant, what is one possible value of $a$?

▶️Answer/Explanation

Ans: 2,4,8

Question

A biologist grows a culture of bacteria as part of an experiment. At the start of the experiment, there are 75 bacteria in the culture. The biologist observes that the population of bacteria doubles every 18 minutes. Which of the following equations best models the number, $n$, of bacteria $t$ hours after the start of the experiment?
A. $n=75(2)^{\frac{t}{18}}$
B. $n=75\left(1+\frac{t}{18}\right)$
C. $n=75(2)^{\frac{10 t}{3}}$
D. $n=75\left(1+\frac{10}{3} t\right)$

▶️Answer/Explanation

Ans: C