f(x)=24(2)^{ {x}{6}} Which table gives four values of x and their corresponding values of f(x) for the given exponential function?

B. x -6 0 6 12 :---: :---: :---: :---: :---: f(x) 12 24 48 96

A. x -6 0 6 12 :---: :---: :---: :---: :---: f(x) 12 0 48 96

C. x -6 0 6 12 :---: :---: :---: :---: :---: f(x) -12 24 48 96

D. x -6 0 6 12 :---: :---: :---: :---: :---: f(x) 12 24 48 72

k(x) = - {1}{2}x^2 + 8x + 20 The function k gives the estimated number of students in a school organization x years after the organization was established, where 0 \le x \le 10. Which statement is the best interpretation of k(4) = 44?

C. It is estimated there were 44 students in the school organization 4 years after it was established.

A. It is estimated there were 4 students in the school organization when it was established.

B. It is estimated there were 44 students in the school organization when it was established.

D. It is estimated there were 44 students in the school organization 10 years after it was established.

Which expression is equivalent to 7 x^{5}-6 x^{4}+9 x^{3}?

D. 6 x^{5} (-6 x^{4}+9 x^{3}+1 )

The growth of the number of bacteria in a certain population is modeled by a function that increases exponentially over time. Which of the following could describe how the number of bacteria in the population changes each hour?

D. Each hour, the number of bacteria in the population is 23\% greater than it was the previous hour.

A. Each hour, the number of bacteria in the population is 23\% less than it was the previous hour.

B. Each hour, the number of bacteria in the population is 2,300 less than it was the previous hour.

C. Each hour, the number of bacteria in the population is 2,300 greater than it was the previous hour.

m(t) = -0.0274 ( {t}{7} )^2 + 7.3873 ( {t}{7} ) + 75.032 The function m gives the predicted body mass m(t), in kilograms (kg), of a male giraffe t days after it was born in a wildlife reserve, where t \le 390. Which of the following is the best interpretation of the statement "m(140) is approximately equal to 212" in this context?

B. The predicted body mass of a male giraffe was approximately 212 kg 140 days after it was born.

A. The predicted body mass of a male giraffe was approximately 140 kg 212 days after it was born.

C. The predicted body mass of a male giraffe was approximately 212 kg {140}{7} days after it was born.

D. The predicted body mass of a male giraffe was approximately {140}{7} kg 212 days after it was born.

Which expression is equivalent to (9 t^{4}-4 t^{2}+5 t-6 )- (6 t^{4}-4 t^{3}+5 t^{2}-6 )?

A. 3 t^{4}+4 t^{3}-9 t^{2}+5 t-12

f(x)=-9 x^{2}+113 x The function f models the depth below sea level, in meters, of a certain seal x minutes after the seal dives into the water. What is the best interpretation of f(2)=190 in this context?

C. 2 minutes after the seal dives into the water, its estimated depth is 190 meters below sea level.

A. 190 minutes after the seal dives into the water, its estimated depth is 2 meters below sea level.

B. The seal reaches an estimated maximum depth of 190 meters below sea level.

D. Approximately 2 minutes after diving into the water, the seal returns to sea level.

A sample of a certain isotope takes 29 years to decay to half its original mass. The function s(t) = 136(0.5)^{ {t}{29}} gives the approximate mass of this isotope, in grams, that remains t years after a 136-gram sample starts to decay. Which statement is the best interpretation of s(58) = 34 in this context?

A. Approximately 34 grams of the sample remains 58 years after the sample starts to decay.

B. The mass of the sample has decreased by approximately 34 grams 58 years after the sample starts to decay.

C. The mass of the sample has decreased by approximately 58 grams 34 years after the sample starts to decay.

D. Approximately 58 grams of the sample remains 34 years after the sample starts to decay.

The function f(x)=236(1.022)^{x} models the value, in dollars, of a certain bank account from 1958 through 1973, where x is the number of years after the end of 1958 . Which of the following is the best interpretation of "f(5) is approximately equal to 263" in this context?

D. The value of the bank account is estimated to be approximately 263 dollars at the end of 1963.

A. The value of the bank account is estimated to increase by approximately 263 dollars every 5 years between the end of 1958 and the end of 1973.

B. The value, in dollars, of the bank account is estimated to be approximately 5 times greater at the end of 1963 than at the end of 1958.

C. The value of the bank account is estimated to be approximately 5 dollars greater at the end of 1963 than at the end of 1958.

SAT Advanced Math Practice Questions

Advanced Math carries about 13 to 15 questions on the digital SAT, the same weight as Algebra. It tests nonlinear functions (quadratics and exponentials), nonlinear equations and systems, and rewriting polynomial and rational expressions into equivalent forms.

The trap in this domain is pattern-matching without reading: an exponential decay model and a quadratic profit model can look similar at a glance but reward different first moves. Work on recognizing the function family first, then the algebra. Each question below shows the full path from setup to answer.

Question 1Nonlinear Functions · easy
$f(x)=24(2)^{\frac{x}{6}}$

Which table gives four values of $x$ and their corresponding values of $f(x)$ for the given exponential function?
A| $x$ | -6 | 0 | 6 | 12 | | :---: | :---: | :---: | :---: | :---: | | $f(x)$ | 12 | 0 | 48 | 96 |
B| $x$ | -6 | 0 | 6 | 12 | | :---: | :---: | :---: | :---: | :---: | | $f(x)$ | 12 | 24 | 48 | 96 |
C| $x$ | -6 | 0 | 6 | 12 | | :---: | :---: | :---: | :---: | :---: | | $f(x)$ | -12 | 24 | 48 | 96 |
D| $x$ | -6 | 0 | 6 | 12 | | :---: | :---: | :---: | :---: | :---: | | $f(x)$ | 12 | 24 | 48 | 72 |
Show answer & explanation
Correct answer: B
| $x$ | -6 | 0 | 6 | 12 | | :---: | :---: | :---: | :---: | :---: | | $f(x)$ | 12 | 24 | 48 | 96 |
Option A is incorrect
This table claims $f(0) = 0$ , but $f(0) = 24(2)^0 = 24$ . An exponential function with a positive starting amount can never output 0 — the graph approaches zero but never touches it.
Option B is correct

Every table uses the same four inputs, $x = -6, 0, 6, 12$ , so compute $f$ at each and match.

At $x = 0$ : $f(0) = 24(2)^0 = 24$ . That single value already eliminates choice A, which claims $f(0) = 0$ — an exponential function with a positive starting amount can never output 0.

At $x = -6$ : $f(-6) = 24(2)^{-1} = 12$ . A negative exponent means divide by 2, not make the output negative — choice C's -12 is the classic sign trap.

At $x = 6$ : $24(2)^1 = 48$ . At $x = 12$ : $24(2)^2 = 96$ , which eliminates choice D's 72 (that would be linear growth of $+24$ per step, not doubling).

Choice B gives $12, 24, 48, 96$ — doubling every 6 units, exactly what the function says.

Answer: B.
Option C is incorrect
This table claims $f(-6) = -12$ . A negative exponent means divide, not negate: $f(-6) = 24(2)^{-1} = 12$ , a positive number.
Option D is incorrect
This table claims $f(12) = 72$ , which is $48 + 24$ — a fixed step of $+24$ , i.e. linear growth. The function doubles every 6 units, so $f(12) = 24(2)^2 = 96$ .
Question 2Nonlinear equations in one variable and systems of equations in two variables · easy
$k(x) = -\frac{1}{2}x^2 + 8x + 20$

The function $k$ gives the estimated number of students in a school organization $x$ years after the organization was established, where $0 \le x \le 10$ . Which statement is the best interpretation of $k(4) = 44$ ?
AIt is estimated there were 4 students in the school organization when it was established.
BIt is estimated there were 44 students in the school organization when it was established.
CIt is estimated there were 44 students in the school organization 4 years after it was established.
DIt is estimated there were 44 students in the school organization 10 years after it was established.
Show answer & explanation
Correct answer: C
It is estimated there were 44 students in the school organization 4 years after it was established.
Option A is incorrect
"When it was established" is $x = 0$ , not $x = 4$ — and no part of $k(4) = 44$ says there were 4 students. The 4 is an input in years, not a student count.
Option B is incorrect
The estimate at establishment is $k(0) = 20$ students, not 44. The 44 belongs to year 4.
Option C is correct

In function notation $k(\text{input}) = \text{output}$ : the input $x$ is years after the organization was established, and the output is the estimated number of students.

So $k(4) = 44$ reads: 4 years after establishment, the organization is estimated to have 44 students.

As a check, the numbers are consistent: $k(4) = -\tfrac{1}{2}(16) + 8(4) + 20 = -8 + 32 + 20 = 44$ .

Choices A and B describe the founding year, which is $x = 0$ (and $k(0) = 20$ , not 44). Choice D swaps in the domain endpoint 10, which has nothing to do with $k(4)$ .

Answer: C.
Option D is incorrect
10 is just the endpoint of the model's domain ( $0 \le x \le 10$ ). The statement $k(4) = 44$ says nothing about year 10.
Question 3Equivalent Expressions · easy
Which expression is equivalent to $7 x^{5}-6 x^{4}+9 x^{3}$ ?
A $x^{4}(7 x-6)$
B $x^{3}\left(7 x^{2}-6 x+9\right)$
C $9 x^{3}\left(7 x^{2}-6 x+1\right)$
D $6 x^{5}\left(-6 x^{4}+9 x^{3}+1\right)$
Show answer & explanation
Correct answer: B
$x^{3}\left(7 x^{2}-6 x+9\right)$
Option A is incorrect
Distributing gives $7x^5 - 6x^4$ — the $9x^3$ term has been dropped entirely.
Option B is correct

All three terms share a factor of $x^3$ (the lowest power present), and the coefficients $7, -6, 9$ share no common factor, so the greatest common factor is exactly $x^3$ .

Pull it out: $7x^5 - 6x^4 + 9x^3 = x^3(7x^2 - 6x + 9)$ .

Verify by distributing: $x^3 \cdot 7x^2 = 7x^5$ , $x^3 \cdot (-6x) = -6x^4$ , $x^3 \cdot 9 = 9x^3$ . ✓

Choice A drops the $9x^3$ term entirely. Choice C factors out $9x^3$ , but $7x^2$ times $9x^3$ would give $63x^5$ , not $7x^5$ . Distributing choice D produces $x^9$ terms that don't exist in the original.

Answer: B.
Option C is incorrect
If you factor $9x^3$ out, the first term inside would have to be $\tfrac{7}{9}x^2$ , because $9x^3 \cdot 7x^2 = 63x^5$ , not $7x^5$ .
Option D is incorrect
Distributing $6x^5$ produces $-36x^9 + 54x^8 + 6x^5$ — terms of degree 9 and 8 that don't exist in the original expression.
Question 4Nonlinear Functions · easy
The growth of the number of bacteria in a certain population is modeled by a function that increases exponentially over time. Which of the following could describe how the number of bacteria in the population changes each hour?
AEach hour, the number of bacteria in the population is 23% less than it was the previous hour.
BEach hour, the number of bacteria in the population is 2,300 less than it was the previous hour.
CEach hour, the number of bacteria in the population is 2,300 greater than it was the previous hour.
DEach hour, the number of bacteria in the population is 23% greater than it was the previous hour.
Show answer & explanation
Correct answer: D
Each hour, the number of bacteria in the population is 23% greater than it was the previous hour.
Option A is incorrect
"23% less each hour" is multiplication by 0.77 every hour — exponential, but decay. The population is described as increasing.
Option B is incorrect
Losing a fixed 2,300 per hour is a constant rate of change: linear decrease, not exponential anything.
Option C is incorrect
Gaining a fixed 2,300 per hour is also a constant rate of change: linear growth. Exponential growth requires the change to be proportional to the current size.
Option D is correct

Exponential growth means the quantity is multiplied by the same factor greater than 1 each hour — equivalently, it changes by a fixed percent of its current size, not by a fixed amount.

"23% greater than the previous hour" means multiplying by 1.23 every hour. That is exponential growth.

Choices B and C describe adding or subtracting a fixed 2,300 each hour — that is a constant rate of change, which is linear, not exponential. Choice A's "23% less" is multiplication by 0.77: exponential, but decay, and the population is growing.

Answer: D.
Question 5Nonlinear equations in one variable and systems of equations in two variables · medium
$m(t) = -0.0274\left(\frac{t}{7}\right)^2 + 7.3873\left(\frac{t}{7}\right) + 75.032$

The function $m$ gives the predicted body mass $m(t)$ , in kilograms (kg), of a male giraffe $t$ days after it was born in a wildlife reserve, where $t \le 390$ . Which of the following is the best interpretation of the statement " $m(140)$ is approximately equal to 212" in this context?
AThe predicted body mass of a male giraffe was approximately 140 kg 212 days after it was born.
BThe predicted body mass of a male giraffe was approximately 212 kg 140 days after it was born.
CThe predicted body mass of a male giraffe was approximately 212 kg $\frac{140}{7}$ days after it was born.
DThe predicted body mass of a male giraffe was approximately $\frac{140}{7}$ kg 212 days after it was born.
Show answer & explanation
Correct answer: B
The predicted body mass of a male giraffe was approximately 212 kg 140 days after it was born.
Option A is incorrect
This swaps input and output: it reads $m(140) \approx 212$ as a 140-kg giraffe at 212 days. The input (inside the parentheses) is days; the output is kilograms.
Option B is correct

The function is defined as: $m(t)$ is the predicted body mass in kilograms $t$ days after birth. Input = days, output = kilograms.

So $m(140) \approx 212$ means: 140 days after birth, the predicted mass is about 212 kg.

Choice A swaps input and output, turning it into a 140-kg giraffe at 212 days. Choices C and D fall for the $\frac{t}{7}$ inside the formula — but that division is already part of how $m$ computes its output. The input to the function is still $t = 140$ days; you do not re-scale the interpretation.

Answer: B.
Option C is incorrect
The $\tfrac{t}{7}$ is part of the formula's internal arithmetic — the function still takes $t$ itself, in days, as its input. $m(140)$ is about day 140, not day $\tfrac{140}{7} = 20$ .
Option D is incorrect
This both swaps input and output and rescales by 7. Neither operation is justified: $m(140) \approx 212$ simply pairs day 140 with mass $\approx 212$ kg.
Question 6Equivalent Expressions · easy
Which expression is equivalent to $\left(9 t^{4}-4 t^{2}+5 t-6\right)-\left(6 t^{4}-4 t^{3}+5 t^{2}-6\right)$ ?
A $3 t^{4}+4 t^{3}-9 t^{2}+5 t-12$
B $3 t^{4}+4 t^{3}-9 t^{2}+5 t$
C $3 t^{4}+10 t-12$
D $3 t^{4}$
Show answer & explanation
Correct answer: B
$3 t^{4}+4 t^{3}-9 t^{2}+5 t$
Option A is incorrect
The -12 constant comes from computing $-6 - 6$ — but subtracting the second parenthesis flips its -6 to $+6$ , so the constants cancel: $-6 + 6 = 0$ .
Option B is correct

Distribute the minus sign through the second parenthesis: $9t^4 - 4t^2 + 5t - 6 - 6t^4 + 4t^3 - 5t^2 + 6$ .

Combine like terms by degree. $t^4$ : $9 - 6 = 3$ , giving $3t^4$ . $t^3$ : only $+4t^3$ . $t^2$ : $-4 - 5 = -9$ , giving $-9t^2$ . $t^1$ : only $+5t$ . Constants: $-6 + 6 = 0$ .

Result: $3t^4 + 4t^3 - 9t^2 + 5t$ .

Choice A is what you get if the constants were $-6 - 6$ — the sign-flip error on the last term. Choices C and D wrongly merge terms of different degrees ( $-4t^2$ with $-5t^2$ is legal; $4t^3$ with anything else is not).

Answer: B.
Option C is incorrect
This merges terms of different degrees: the $4t^3$ and $-9t^2$ terms have been absorbed into others. Only terms with the same power can combine.
Option D is incorrect
This keeps only the $t^4$ terms and discards $4t^3$ , $-9t^2$ , and $5t$ , none of which cancel.
Question 7Nonlinear Functions · easy
$f(x)=-9 x^{2}+113 x$

The function $f$ models the depth below sea level, in meters, of a certain seal $x$ minutes after the seal dives into the water. What is the best interpretation of $f(2)=190$ in this context?
A190 minutes after the seal dives into the water, its estimated depth is 2 meters below sea level.
BThe seal reaches an estimated maximum depth of 190 meters below sea level.
C2 minutes after the seal dives into the water, its estimated depth is 190 meters below sea level.
DApproximately 2 minutes after diving into the water, the seal returns to sea level.
Show answer & explanation
Correct answer: C
2 minutes after the seal dives into the water, its estimated depth is 190 meters below sea level.
Option A is incorrect
This swaps input and output. The input 2 is minutes and the output 190 is meters, not the other way around.
Option B is incorrect
190 is just the depth at the 2-minute mark, not the maximum. The parabola's vertex is at $x = \tfrac{113}{18} \approx 6.3$ minutes, where the depth is about 355 meters.
Option C is correct

By the function's definition, the input $x$ is minutes after the dive and the output $f(x)$ is depth below sea level in meters.

So $f(2) = 190$ says: 2 minutes after diving, the seal's estimated depth is 190 meters below sea level. Check: $f(2) = -9(4) + 113(2) = -36 + 226 = 190$ . ✓

Choice A swaps input and output. Choice B claims 190 is the maximum depth, but the vertex of this parabola is at $x = \frac{113}{18} \approx 6.3$ minutes, where the depth is about 355 meters — 190 is just the depth at one moment. Choice D describes returning to sea level, which is $f(x) = 0$ , not 190.

Answer: C.
Option D is incorrect
Returning to sea level means depth 0, i.e. solving $f(x) = 0$ — which happens at $x \approx 12.6$ minutes, not 2. $f(2) = 190$ says the seal is 190 meters down.
Question 8Nonlinear equations in one variable and systems of equations in two variables · easy
A sample of a certain isotope takes 29 years to decay to half its original mass. The function $s(t) = 136(0.5)^{\frac{t}{29}}$ gives the approximate mass of this isotope, in grams, that remains $t$ years after a 136-gram sample starts to decay. Which statement is the best interpretation of $s(58) = 34$ in this context?
AApproximately 34 grams of the sample remains 58 years after the sample starts to decay.
BThe mass of the sample has decreased by approximately 34 grams 58 years after the sample starts to decay.
CThe mass of the sample has decreased by approximately 58 grams 34 years after the sample starts to decay.
DApproximately 58 grams of the sample remains 34 years after the sample starts to decay.
Show answer & explanation
Correct answer: A
Approximately 34 grams of the sample remains 58 years after the sample starts to decay.
Option A is correct

The function's definition fixes the meaning: $s(t)$ is the mass in grams that remains $t$ years after decay starts. Input = years elapsed, output = grams remaining.

So $s(58) = 34$ means: 58 years in, about 34 grams remain.

The numbers agree with the half-life story: 58 years is two half-lives of 29 years, and $136 \to 68 \to 34$. ✓

Choice B confuses "remains" with "has decreased by" — the decrease is $136 - 34 = 102$ grams, not 34. Choices C and D swap the roles of 34 and 58.

Answer: A.
Option B is incorrect
$s(t)$ measures what remains, not what was lost. The decrease after 58 years is $136 - 34 = 102$ grams.
Option C is incorrect
This swaps the numbers and misreads "remains" as "decreased": neither 58 grams of loss nor a 34-year mark appears in $s(58) = 34$ .
Option D is incorrect
This swaps input and output: 58 is the elapsed time in years (two half-lives), and 34 is the grams remaining — not the reverse.
Question 9Equivalent Expressions · easy
Which expression is equivalent to $(5x^3 - 3x + 8) - (9x^6 + 5x - 3)$ ?
A $-4x^9 - 8x - 11$
B $-9x^6 + 5x^3 - 8x + 11$
C $-9x^6 + 5x^3 + 2x + 5$
D $-4x^9 - 2x + 5$
Show answer & explanation
Correct answer: B
$-9x^6 + 5x^3 - 8x + 11$
Option A is incorrect
The $x^9$ comes from multiplying $x^3 \cdot x^6$ , but this is subtraction — unlike terms stay separate. The signs are also mangled.
Option B is correct

Distribute the minus sign: $5x^3 - 3x + 8 - 9x^6 - 5x + 3$ .

Combine like terms. $x^6$ : only $-9x^6$ . $x^3$ : only $5x^3$ . $x$ : $-3 - 5 = -8$ , giving $-8x$ . Constants: $8 + 3 = 11$ .

Written in descending degree: $-9x^6 + 5x^3 - 8x + 11$ .

Choices A and D invent $x^9$ by multiplying $x^3 \cdot x^6$ — but this is subtraction, not multiplication, and unlike terms can only sit side by side. Choice C forgets to flip the signs of $+5x$ and -3 when distributing the minus.

Answer: B.
Option C is incorrect
The $+2x$ and $+5$ come from forgetting to flip the signs of $+5x$ and -3 when distributing the minus: correctly, $-3x - 5x = -8x$ and $8 + 3 = 11$ .
Option D is incorrect
Again an invented $x^9$ from multiplying exponents of unlike terms; subtraction can only combine terms of identical degree.
Question 10Nonlinear Functions · medium
The function $f(x)=236(1.022)^{x}$ models the value, in dollars, of a certain bank account from 1958 through 1973, where $x$ is the number of years after the end of 1958 . Which of the following is the best interpretation of " $f(5)$ is approximately equal to 263" in this context?
AThe value of the bank account is estimated to increase by approximately 263 dollars every 5 years between the end of 1958 and the end of 1973.
BThe value, in dollars, of the bank account is estimated to be approximately 5 times greater at the end of 1963 than at the end of 1958.
CThe value of the bank account is estimated to be approximately 5 dollars greater at the end of 1963 than at the end of 1958.
DThe value of the bank account is estimated to be approximately 263 dollars at the end of 1963.
Show answer & explanation
Correct answer: D
The value of the bank account is estimated to be approximately 263 dollars at the end of 1963.
Option A is incorrect
$f(5) \approx 263$ is the account's total value at one moment, not a recurring gain. The actual growth over those 5 years is $263 - 236 = 27$ dollars.
Option B is incorrect
The account grows by a factor of $(1.022)^5 \approx 1.11$ over 5 years — about 11%, nowhere near "5 times greater". The 5 is the elapsed time, not a growth factor.
Option C is incorrect
The increase is $263 - 236 \approx 27$ dollars, not 5. The 5 in $f(5)$ counts years after the end of 1958.
Option D is correct

The input $x$ counts years after the end of 1958, and the output is the account's value in dollars. So $f(5)$ is the value 5 years later — at the end of 1963.

$f(5) \approx 263$ therefore means: the account is worth about 263$ at the end of 1963.

Choice A turns a one-time value into a recurring increase — the actual growth from 236 $to$ \263 over those 5 years is about 27 $, not$ \263. Choice B misreads the output 263 as "5 times greater." Choice C misreads it as a 5$ increase.

Answer: D.

Want the other 4,090+ questions?

These 10 questions are a public sample. BrightSAT has 4,100+ digital SAT questions across every domain and skill, plus Bluebook-style full-length adaptive practice tests, targeted practice, and score reports that show exactly where you lose points.

Start practicing free