20 medium SAT Nonlinear functions questions

Medium is where most scores are actually won and lost. These questions are not tricky for the sake of it, but every one of them has a wrong answer built to catch a specific shortcut.

Every question below is a real item from the SAT Climb bank, tagged medium by the same difficulty model the app uses to build your practice. Pick an answer before you open the explanation.

Math · Advanced Math~5 per testMedium tier

None of these repeat the 25 hard Nonlinear functions questions, so the two pages together are 45 different questions on this one skill.

Question 1Medium

The function f is defined by $f(x) = (x - 3)^{2} + 7$ . What is the minimum value of f(x)?

Show the answer and explanation

Why B is right

The function $f(x) = (x - 3)^{2} + 7$ is in vertex form. The vertex of a quadratic in the form $(x - h)^{2} + k$ is at (h, k). Since the coefficient of the squared term is positive, the parabola opens upward, and the vertex represents the minimum value. Therefore, the minimum value is $k = 7$ .

Why the others are wrong

AThis is the x-coordinate of the vertex, not the minimum value of the function.
CThis incorrectly applies a sign error to the k-value from the vertex form.
DThis incorrectly doubles the minimum value, possibly confusing the transformation with the coefficient.

Question 2Medium

A population of bacteria starts at 500 and doubles every 3 hours. Which function models the number of bacteria, b, after h hours?

Show the answer and explanation

Why A is right

The population doubles every 3 hours, so the growth factor is 2 applied every 3 hours. After h hours, the number of 3-hour periods is $h/3$ , giving $b(h) = 500$ (2)^ $(h/3)$ .

Why the others are wrong

BThis incorrectly places h in the exponent as 3h instead of h/3, reversing the relationship between time and doubling periods.
CThis swaps the base and the period values, confusing 2 (the growth factor) with 3 (the time period).
DThis incorrectly models exponential growth as linear growth, dropping the exponential term entirely.

Question 3Medium

The function f is defined by $f(x) = x^{2} + 4x - 5$ . If the graph of $y = f$ (x) in the xy-plane is shifted up 9 units, what is the y-intercept of the resulting graph?

Show the answer and explanation

Why B is right

Shifting the graph up 9 units gives the new function $g(x) = f (x) + 9$ = $x^{2}$ + $4x - 5 + 9$ = $x^{2}$ + $4x + 4$ . The y-intercept occurs at $x = 0$ , so g(0) = 0 + 0 + 4 = 4. The y-intercept is (0, 4).

Why the others are wrong

AThis incorrectly uses the original y-intercept without applying the upward shift.
CThis incorrectly uses only the shift amount as the new y-intercept.
DThis incorrectly adds the shift to the original y-intercept with a sign error.

Question 4Medium

A chemist models the concentration of a compound in a solution using the function C, where

C(h) = 85(0.75)^{h}

gives the concentration in milligrams per liter h hours after the reaction begins.

What is the best interpretation of the statement "C(6) is approximately equal to 15" in this context?

Show the answer and explanation

Why C is right

The function C(h) gives the concentration in milligrams per liter after h hours. The statement C(6) ≈ 15 means when $h = 6$ , the concentration C is approximately 15. Therefore, 6 hours after the reaction begins, the concentration is approximately 15 milligrams per liter.

Why the others are wrong

AThis reverses the input and output variables, treating the concentration as the time and the time as the concentration.
BThis misinterprets the function value as a rate of decrease rather than the actual concentration at a specific time.
DThis multiplies the correct value by 10, misinterpreting the scale of the output.

Question 5Medium

The function f is defined by $f(x) = x^{2} + 4x - 12$ . What is the product of the x-intercepts of the graph of $y = f$ (x) in the xy-plane?

Show the answer and explanation

Why D is right

By Vieta's formulas, for a quadratic $x^{2}$ + bx + c, the product of the roots (x-intercepts) equals c. Here $c = -12$ , so the product of the x-intercepts is -12. Alternatively, factoring $f(x) = (x + 6) (x - 2)$ gives roots -6 and 2, whose product is (-6)(2) = -12.

Why the others are wrong

AThis is the coefficient of x, which gives the sum of the x-intercepts, not their product.
BThis results from using the absolute value of the constant term without considering its sign.
CThis results from dividing the coefficient of x by 2, confusing formulas for vertex and intercepts.

Question 6Medium

The function f is defined by $f(x) = 4x^{2} - 16x + k$ , where k is a constant. If the minimum value of f(x) is 3, what is the value of k?

Show the answer and explanation

Why C is right

The x-coordinate of the vertex is at $x = -(-16)/(2\cdot 4) = 16/8 = 2$ . The minimum value occurs at this x-coordinate, so $f(2) = 3$ . Substituting: $4(2)^{2} - 16 (2) + k = 3$ , which gives $16 - 32 + k = 3$ , so $k = 19$ .

Why the others are wrong

AThis uses the minimum value directly as k without accounting for the other terms at the vertex.
BThis uses the absolute value of the coefficient of x without completing the calculation.
DThis results from an arithmetic error in computing 16 - 32 + k = 3.

Question 7Medium

The function f is defined by $f(x) = x^{2} + 8x + 12$ . What is the value of x when $f(x) = 0$ ?

Show the answer and explanation

Why C is right

Setting $f(x) = 0$ gives $x^{2}$ + $8x + 12 = 0$ . Factoring yields $(x + 6) (x + 2) = 0$ , so $x = -6$ or $x = -2$ . Since -2 is one of the answer choices, it is a correct value.

Why the others are wrong

AThis is the x-coordinate of the vertex, not an x-intercept of the parabola.
BThis is the positive of one x-intercept, resulting from a sign error in solving the equation.
DThis is the constant term from the original equation, not an x-intercept.

Question 8Medium

The function f is defined by $f(x) = 3x^{2} + 12x - 15$ . What is the minimum value of f(x)?

Show the answer and explanation

Why A is right

The x-coordinate of the vertex is $x = -12/(2\cdot 3) = -2$ . Substituting $x = -2$ into f(x) gives f(-2) = $3(-2)^{2} + 12$ (-2) - 15 = 3(4) - 24 - 15 = 12 - 24 - 15 = -27. Since $a = 3$ is positive, the parabola opens upward and this is the minimum value.

Why the others are wrong

BThis incorrectly uses the constant term -15 as the minimum value without evaluating the function at the vertex.
CThis uses only the x-coordinate of the vertex (-2) instead of computing the y-coordinate by evaluating f(-2).
DThis results from a sign error in computing f(-2), getting positive 27 instead of -27.

Question 9Medium

The function f is defined by f(x) = - $3(x + 2)^{2} + 12$ . In the xy-plane, the graph of $y = f$ (x) is translated 5 units to the right to produce the graph of $y = g$ (x). Which equation defines function g?

Show the answer and explanation

Why A is right

Translating a graph 5 units to the right replaces x with $(x - 5)$ . Starting with $f(x) = -3 (x + 2)^{2} + 12$ , we get $g(x) = -3 ((x - 5) + 2)^{2} + 12 = -3 (x - 3)^{2} + 12$ .

Why the others are wrong

BThis results from incorrectly adding 5 to the h-value instead of subtracting, translating left instead of right.
CThis results from incorrectly translating vertically by adding 5 to the k-value instead of horizontally.
DThis results from replacing (x + 2) with (x - 5) instead of correctly applying the translation to get (x - 3).

Question 10Medium

A geologist models the temperature of a cooling lava flow using the function T, where

T(m) = 1200(0.91)^{m}

gives the temperature in degrees Celsius m minutes after initial measurement.

What is the best interpretation of the statement "T(20) is approximately equal to 185" in this context?

Show the answer and explanation

Why A is right

The function T(m) gives the temperature in degrees Celsius after m minutes. The statement T(20) ≈ 185 means when $m = 20$ , the temperature T is approximately 185. Therefore, 20 minutes after initial measurement, the temperature is approximately 185 degrees Celsius.

Why the others are wrong

BThis reverses the input and output, treating the temperature as the time variable and time as the temperature.
CThis multiplies the correct temperature by 10, misinterpreting the scale of the output.
DThis incorrectly interprets the function value as a rate of cooling rather than the actual temperature at a specific time.

Question 11Medium

The function f is defined by $f(x) = x^{2} + 2x - 15$ . What is the sum of the x-intercepts of the graph of $y = f$ (x) in the xy-plane?

Show the answer and explanation

Why B is right

For a quadratic $ax^{2}$ + bx + c, the sum of the roots is $-b/a$ . Here, $a = 1$ and $b = 2$ , so the sum is -2/1 = -2. Alternatively, factoring gives $(x + 5) (x - 3) = 0$ , so roots are -5 and 3, which sum to -2.

Why the others are wrong

AThis incorrectly uses the constant term as the sum of the x-intercepts.
CThis results from a sign error in applying the sum-of-roots formula, computing 2 instead of -2.
DThis incorrectly uses the absolute value of the constant term as the sum.

Question 12Medium

The function f is defined by $f(x) = 2(x - 5)^{2} + 3$ . For what value of x does $f(x) = 21$ ?

Show the answer and explanation

Why B is right

Setting $f(x) = 21$ : $2(x - 5)^{2} + 3 = 21$ , so $2(x - 5)^{2} = 18$ , $(x - 5)^{2} = 9$ , $x - 5$ = $\pm 3$ . This gives $x = 8$ or $x = 2$ . The answer 8 is provided.

Why the others are wrong

AThis results from solving (x - 5)² = 9 by taking only the square root of 9 and not adding it back to 5 correctly.
CThis results from a sign error when solving for x.
DThis results from incorrectly computing the square root step.

Question 13Medium

The function f is defined by $f(x) = 4(x - 3)^{2} - 64$ . What is the minimum value of f(x)?

Show the answer and explanation

Why A is right

The function is in vertex form $f(x) = a (x - h)^{2} + k$ with $a = 4$ , $h = 3$ , and $k = -64$ . Since $a = 4$ is positive, the parabola opens upward and the vertex represents the minimum. The minimum value is $k = -64$ .

Why the others are wrong

BThis results from dividing k by the coefficient a, computing -64/4 = -16, incorrectly applying the coefficient to the constant term.
CThis incorrectly uses the x-coordinate of the vertex (h = 3) as the minimum value instead of the y-coordinate.
DThis results from a sign error, using positive 64 instead of -64 from the vertex form.

Question 14Medium

The function f is defined by $f(x) = x^{2} + 8x + 15$ . In the xy-plane, the graph of $y = h$ (x) is the result of shifting the graph of $y = f$ (x) left 4 units. Which equation defines function h?

Show the answer and explanation

Why A is right

Shifting left 4 units means replacing x with $(x + 4)$ . So $h(x) = f (x + 4) = (x + 4)^{2} + 8 (x + 4) + 15$ = $x^{2}$ + $8x + 16 + 8x + 32 + 15$ = $x^{2}$ + $16x + 63$ .

Why the others are wrong

BThis incorrectly shifts the constant term instead of substituting (x + 4) into the function.
CThis incorrectly adjusts only the coefficient of x without properly expanding f(x + 4).
DThis incorrectly shifts by adding to the constant term and dropping the x term.

Question 15Medium

A meteorologist models atmospheric pressure using the function P, where

P(a) = 1013(0.88)^{a}

gives the pressure in millibars at an altitude of a kilometers above sea level.

What is the best interpretation of the statement "P(8) is approximately equal to 310" in this context?

Show the answer and explanation

Why D is right

The function P(a) gives the pressure in millibars at an altitude of a kilometers. The statement P(8) ≈ 310 means when $a = 8$ , the pressure P is approximately 310. Therefore, at an altitude of 8 kilometers above sea level, the pressure is approximately 310 millibars.

Why the others are wrong

AThis reverses the input and output, treating the pressure as the altitude and the altitude as the pressure.
BThis incorrectly interprets the function value as a rate of change rather than the actual pressure at a specific altitude.
CThis multiplies the correct pressure by 10, misinterpreting the scale of the output.

Question 16Medium

The function f is defined by $f(x) = x^{2} - 14x + 45$ . What is the product of the x-intercepts of the graph of $y = f$ (x) in the xy-plane?

Show the answer and explanation

Why D is right

For a quadratic f(x) = $x^{2}$ + bx + c, the product of the roots equals c. Here, $c = 45$ , so the product of the x-intercepts is 45. Alternatively, factoring gives $f(x) = (x - 5) (x - 9)$ , and 5 × 9 = 45.

Why the others are wrong

AThis confuses the coefficient of x with the product of the x-intercepts.
BThis results from confusing the sum of the x-intercepts with their product, and using the wrong sign.
CThis results from incorrectly determining the sign when identifying the constant term as the product.

Question 17Medium

The function f is defined by f(x) = - $(x - 1)^{2} + 25$ . For what positive value of x does $f(x) = 0$ ?

Show the answer and explanation

Why C is right

Setting $f(x) = 0$ gives - $(x - 1)^{2} + 25 = 0$ , which simplifies to $(x - 1)^{2} = 25$ . Taking the square root of both sides yields $x - 1$ = $\pm 5$ , so $x = 1 + 5 = 6$ or $x = 1 - 5 = -4$ . The positive value is $x = 6$ .

Why the others are wrong

AThis incorrectly subtracts 1 from the square root value of 25.
BThis uses the square root of 25 directly without considering the horizontal shift.
DThis is the negative solution to the equation, but the question asks for the positive value.

Question 18Medium

The function f is defined by $f(x) = x^{2} - 10x + 21$ . What is the sum of the x-intercepts of the graph of $y = f$ (x) in the xy-plane?

Show the answer and explanation

Why C is right

Factoring: $f(x) = (x - 3) (x - 7)$ . The x-intercepts are at $x = 3$ and $x = 7$ . Their sum is 3 + 7 = 10. Alternatively, by Vieta's formulas for $x^{2}$ - $10x + 21$ , the sum of roots equals the negative of the coefficient of x, which is 10.

Why the others are wrong

AThis is the constant term, which equals the product of the x-intercepts, not their sum.
BThis results from using the coefficient of x directly without changing its sign.
DThis results from dividing the coefficient of x by 2, confusing the sum of roots with the x-coordinate of the vertex.

Question 19Medium

The function f is defined by $f(x) = x^{2} + 8x + 12$ . If the graph of $y = f$ (x) in the xy-plane is translated 3 units to the right to form the graph of $y = g$ (x), what is the minimum value of g(x)?

Show the answer and explanation

Why D is right

First, complete the square for f(x): $f(x) = (x + 4)^{2} - 4$ , so the minimum is -4 at $x = -4$ . Translating 3 units right shifts the vertex to $x = -1$ , but the minimum value remains -4.

Why the others are wrong

AThis incorrectly subtracts 3 from the original minimum value.
BThis incorrectly identifies the new x-coordinate of the vertex as the minimum value.
CThis results from a sign error when completing the square.

Question 20Medium

An environmental engineer models the amount of pollutant in a water treatment system using the function A, where

A(d) = 950(0.82)^{d}

gives the amount in grams d days after treatment begins.

What is the best interpretation of the statement "A(14) is approximately equal to 95" in this context?

Show the answer and explanation

Why D is right

The function A(d) gives the amount of pollutant in grams after d days. The statement A(14) ≈ 95 means when $d = 14$ , the amount A is approximately 95. Therefore, 14 days after treatment begins, the amount of pollutant is approximately 95 grams.

Why the others are wrong

AThis incorrectly interprets the function value as a rate of decrease per period rather than the actual amount at a specific time.
BThis uses the initial amount from the function rather than the calculated value at d = 14, ignoring the exponential decay.
CThis reverses the input and output, treating the amount as the time variable and time as the amount.

What to do after medium

Medium is the tier that decides most scores. If these are landing, the hard set is where the remaining points are.

Easy Nonlinear functions questions Hard Nonlinear functions questions Start free →

Questions are written by SAT Climb and drawn from its own item bank. SAT® is a registered trademark of College Board, which is not affiliated with and does not endorse SAT Climb.

Why B is right

Why the others are wrong

Why A is right

Why the others are wrong

Why B is right

Why the others are wrong

Why C is right

Why the others are wrong

Why D is right

Why the others are wrong

Why C is right

Why the others are wrong

Why C is right

Why the others are wrong

Why A is right

Why the others are wrong

Why A is right

Why the others are wrong

Why A is right

Why the others are wrong

Why B is right

Why the others are wrong

Why B is right

Why the others are wrong

Why A is right

Why the others are wrong

Why A is right

Why the others are wrong

Why D is right

Why the others are wrong

Why D is right

Why the others are wrong

Why C is right

Why the others are wrong

Why C is right

Why the others are wrong

Why D is right

Why the others are wrong

Why D is right

Why the others are wrong

What to do after medium

More on Nonlinear functions