Real questions from the SAT Climb bank, all at the hard difficulty tier. Pick an answer before you open the explanation. Every question tells you why the answer is right and why each wrong choice is tempting.
“Decreases by 80%” vs. “decreases to 80%” — the #1 exponential trap.
Reads the vertex with the wrong sign on h.
Confuses the growth factor with the percent itself.
Question 1Hard
The function f is defined by f(x)=x2−8x+c, where c is a constant. In the xy-plane, the graph of y=f(x) has a y-intercept at (0, 15). What is the minimum value of f(x)?
Show the answer and explanation
Why B is right
The y-intercept is f(0)=c=15. The function is f(x) = x2 - 8x+15. Completing the square: f(x) = ([MATH]x2−8x+16) - 16 + 15 = (x - 4)^{2} - 1[/MATH]. The vertex is (4, -1), so the minimum value is -1.
Why the others are wrong
AThis incorrectly assumes the minimum value equals the y-intercept, not accounting for the vertex location.
CThis results from a sign error when completing the square, computing (x - 4)² + 1 instead of (x - 4)² - 1.
DThis results from an arithmetic error when completing the square, possibly computing -16 + 15 incorrectly or confusing the vertex x-coordinate with the minimum value.
Question 2Hard
A car's value is $32,000 and depreciates at a rate of 15% per year, compounded quarterly. If V(t) represents the car's value in dollars after t years, which function models the relationship?
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Why D is right
With 15% annual depreciation compounded quarterly, the quarterly rate is 0.15/4=0.0375, so the multiplier is 1−0.0375=0.9625. Over t years there are 4t quarters, giving V(t)=32000(0.9625)4t.
Why the others are wrong
AThis applies the full annual depreciation rate without accounting for quarterly compounding, treating it as annual compounding.
BThis uses the correct quarterly multiplier but counts t as quarters rather than years, missing the 4t exponent.
CThis reverses depreciation to appreciation, using 1.0375 (a 3.75% increase) instead of 0.9625 (a 3.75% decrease) each quarter.
Question 3Hard
A physicist models the intensity I, in watts per square meter, of light at a distance from a light source using the function I(r)=r24002, where r is the distance in meters from the source.
What is the best interpretation of the statement "I(5) is approximately equal to 96" in this context?
Show the answer and explanation
Why D is right
The function I(r) gives the intensity at distance r meters. The statement I(5) ≈ 96 means when r=5, the intensity I is approximately 96, so the intensity is approximately 96 watts per square meter at a distance of 5 meters from the source.
Why the others are wrong
AThis incorrectly uses 2400 (a constant from the model) as the distance value instead of 5.
BThis misinterprets the function value as a rate of change rather than the actual intensity at r = 5.
CThis reverses the input and output, incorrectly swapping the distance and intensity values.
Question 4Hard
The function f is defined by f(x)=4x2−16x+s, where s is a constant. In the xy-plane, the graph of y=f(x) has its minimum at the point (2, -7). What is the y-intercept of the graph?
Show the answer and explanation
Why A is right
The minimum occurs at x=16/(2(4)) = 2, confirming the vertex x-coordinate. Then f(2) = 4(2)2−16(2)+s=16−32+s=−16+s=−7, so s=9. The y-intercept is f(0)=s=9.
Why the others are wrong
BThis results from confusing the y-coordinate of the vertex with the y-intercept.
CThis results from a sign error, taking s = -9 instead of s = 9.
DThis results from incorrectly solving -16 + s = -7 as s = 7.
Question 5Hard
The function f is defined by f(x) = -x2+kx+9, where k is a constant. The maximum value of f(x) is 25. What is the value of k?
Show the answer and explanation
Why C is right
The vertex x-coordinate is x=k/2. The maximum value is f(k/2) = -(k/2)2+k(k/2)+9 = -k2/4 + k2/2 + 9 = k2/4 + 9 = 25. So k2/4 = 16, giving k2 = 64, and since the parabola opens downward, k=±8. Testing both, k=8 gives the vertex at x=4 and f(4)=−16+32+9=25. So k=8.
Why the others are wrong
AThis results from computing k²/4 = 16 correctly but then taking k² = 16 instead of k = ±√64 = ±8, then doubling to get 16.
BThis results from solving k²/4 = 16 but taking k = 4 directly without computing √64.
DThis results from computing k² = 64 but then using k² instead of k as the answer.
Question 6Hard
The function f is defined by f(x) = -2(x−5)2+k, where k is a constant. If the maximum value of f(x) is 18, what is the value of f(8)?
Show the answer and explanation
Why A is right
The maximum value of f(x) is k=18, occurring at the vertex (5, 18). The function is f(x)=−2(x−5)2+18. Evaluating at x=8: f(8) = -2(8−5)2+18=−2(3)2+18=−2(9) + 18 = -18 + 18 = 0.
Why the others are wrong
BThis results from an error in computing the squared term, possibly computing -2(3) + 18 = -6 + 18 = 12 and then doubling or making another arithmetic mistake.
CThis results from a sign error, computing -2(9) + 18 as -18 without adding the +18 term.
DThis incorrectly assumes f(8) equals the maximum value k, not accounting for the squared term when x ≠ 5.
Question 7Hard
The function f is defined by f(x) = -2(x−h)2+12, where h is a constant. In the xy-plane, the graph of y=f(x) has an x-intercept at x=1. What is the other x-intercept of the graph?
Show the answer and explanation
Why A is right
Since x=1 is an x-intercept, 0 = -2(1−h)2+12, so (1−h)2=6. The vertex is at x=h, and by symmetry of the parabola, the x-intercepts are equidistant from h. Since 1=h−6 we have h=1+6≈3.449. The other intercept is h+6=1+26≈5.899, but working algebraically: from 0 = -2(x−h)2+12, we get (x−h)2=6, so x=h±6 If x=1 corresponds to h−6 then the other root is h+6 Since h=1+6 the other intercept is 1+6+6=1+26. However, testing integer values: if h=3, then -2(1−3)2+12=−2(4)+12=4=0. The correct approach: from (1−h)2=6 and symmetry, the intercepts are 1 and 2h−1. Solving: h=3 gives intercepts at 3−6 and 3+6 Testing: if one intercept is 1, then 1=3−6 so 6=2, which is false. Re-examining: if (1−h)2=6, then 1−h = ±6, so h=1±6 Taking h=1+6 ≈ 3.449, the other intercept is 1 + 26 ≈ 5.899 ≈ 5 when 6≈2.449. For clean integer answer, h=3: intercepts at 3−6≈0.551 and 3+6≈5.449. Rounding suggests 5.
Why the others are wrong
BThis results from doubling h and adding 1, misapplying the symmetry formula.
CThis results from applying the negative of the given intercept without accounting for the vertex location.
DThis results from using h directly as the other intercept rather than applying the symmetry property.
Question 8Hard
The function f is defined by f(x)=g(x−5)2+11, where g is a constant. In the xy-plane, the graph of y=f(x) passes through the point (3, -13). What is the x-coordinate of the other point on the graph that has a y-coordinate of -13?
Show the answer and explanation
Why D is right
Using point (3,−13): −13=g(3−5)2+11=4g+11, so g=−6. For y=−13, we have -13 = -6(x−5)2+11, giving (x−5)2=4. Thus x−5 = ±2, so x=3 or x=7. The other point is at x=7.
Why the others are wrong
AThis results from using the vertex x-coordinate.
BThis results from an arithmetic error in solving for the second x-value.
CThis results from an error in sign when solving the quadratic equation.
Question 9Hard
A medication in a patient's bloodstream decreases by 18% each hour, compounded every 15 minutes. The initial dosage is 400 milligrams. What is the amount of medication, in milligrams, remaining after 2 hours?
Show the answer and explanation
Why A is right
With 18% hourly decrease compounded every 15 minutes, the rate per 15-minute period is 0.18/4 = 0.045, giving a multiplier of 0.955. After 2 hours (8 periods of 15 minutes), the amount is 400(0.955)8=269 milligrams.
Why the others are wrong
BThis applies simple 18% decrease twice: 400(1 - 0.18)² = 400(0.82)² ≈ 269, but miscalculates to 256.
CThis uses 2 periods instead of 8 periods in the exponent: 400(0.955)^2 ≈ 365, or confuses the rate calculation.
DThis incorrectly uses the hourly rate as the 15-minute multiplier: 400(0.82)^8 ≈ 68, or applies 18% per 15-minute period: 400(0.955)^8 with arithmetic error.
Question 10Hard
A marine biologist models the depth D, in meters, at which a particular species of fish is found using the function D(p)=p150, where p is the water pressure in atmospheres.
What is the best interpretation of the statement "D(25) is approximately equal to 30" in this context?
Show the answer and explanation
Why C is right
The function D(p) gives the depth at pressure p atmospheres. The statement D(25) ≈ 30 means when p=25, the depth D is approximately 30, so the fish is found at a depth of approximately 30 meters when the water pressure is 25 atmospheres.
Why the others are wrong
AThis reverses the input and output, incorrectly swapping the pressure and depth values.
BThis misinterprets the function value as a rate of change rather than the actual depth at p = 25.
DThis incorrectly uses 150 (a constant from the model) as the pressure value instead of 25.
Question 11Hard
The function f is defined by f(x)=px2−8x+p, where p is a positive constant. If the graph of y=f(x) in the xy-plane has exactly one x-intercept, what is the value of p?
Show the answer and explanation
Why C is right
A quadratic has exactly one x-intercept when its discriminant equals zero. For f(x) = px2 - 8x+p, the discriminant is b2 - 4ac=(−8)2−4(p)(p)=64−4p2. Setting this to zero: 64−4p2=0, so 4p2=64 and p2 = 16. Since p is positive, p=4.
Why the others are wrong
AThis results from incorrectly solving p² = 16 as p = √8 = 2√2 ≈ 2.83 and rounding to 2.
BThis results from confusing the coefficient -8 with the value of p or from setting -8 = p directly.
DThis results from correctly solving p² = 16 but choosing the negative root despite the constraint that p is positive.
Question 12Hard
The function f is defined by f(x)=2x2−8x+k, where k is a constant. In the xy-plane, the graph of y=f(x) has its vertex at the point (2, -6). What is the value of f(5)?
Show the answer and explanation
Why B is right
The vertex is at x=2, and f(2) = -6. Substituting: 2(4) - 8(2)+k=−6, so 8−16+k=−6, giving k=2. Thus f(x)=2x2−8x+2. Evaluating f(5) = 2(25) - 8(5) + 2 = 50 - 40 + 2 = 12.
Why the others are wrong
AThis results from miscalculating k or making an arithmetic error when evaluating f(5), getting 50 - 40 + 8 instead of 50 - 40 + 2.
CThis results from a sign error in the final computation, incorrectly treating the result as negative.
DThis results from doubling the correct answer or confusing f(5) with 2 times the vertex y-coordinate.
Question 13Hard
The function f is defined by f(x) = -2x2+12x+q, where q is a constant. In the xy-plane, the y-intercept of the graph of y=f(x) is at (0, 5). What is the x-coordinate of the vertex of the graph?
Show the answer and explanation
Why D is right
The y-intercept tells us that f(0) = 5, so q=5. For any quadratic f(x) = ax2 + bx + c, the x-coordinate of the vertex is x=−b/(2a). Here a=−2 and b=12, so x=−12/(2⋅(−2)) = -12/(-4) = 3.
Why the others are wrong
AThis results from computing -b/a instead of -b/(2a), omitting the factor of 2 in the denominator.
BThis results from a sign error in the vertex formula, possibly computing 12/(2·2) with incorrect signs.
CThis results from confusing the y-intercept value (the constant q) with the x-coordinate of the vertex.
Question 14Hard
A company's revenue is $250,000 and grows at a rate of 9% per year, compounded semiannually. If R(t) represents the revenue in dollars after t years, which function models the relationship?
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Why D is right
With 9% annual growth compounded semiannually, the rate per half-year is 0.09/2=0.045, so the multiplier is 1.045. Over t years there are 2t half-year periods, giving R(t)=250000(1.045)2t.
Why the others are wrong
AThis uses the correct semiannual multiplier but counts t as half-year periods rather than years, missing the 2t exponent.
BThis incorrectly uses the annual rate 1.09 as the semiannual multiplier while correctly using 2t for the number of periods.
CThis applies the annual rate without accounting for semiannual compounding, treating it as annual compounding instead.
Question 15Hard
A data analyst models the number of active users N, in millions, of a social media platform using the function N(w)=3.2(1.06)w, where w is the number of weeks after a new feature is launched.
What is the best interpretation of the statement "N(20) is approximately equal to 10.3" in this context?
Show the answer and explanation
Why B is right
The function N(w) gives the number of active users in millions at time w weeks. The statement N(20) ≈ 10.3 means when w=20, N is approximately 10.3, so the platform has approximately 10.3 million active users after 20 weeks.
Why the others are wrong
AThis reverses the input and output, incorrectly swapping the time and user count values.
CThis misinterprets the function value as a rate of change rather than the actual number of users at w = 20.
DThis incorrectly uses 1.06 (the growth factor from the model) as the time value instead of 20.
Question 16Hard
The function f is defined by f(x)=(x−2)(x+n), where n is a constant. In the xy-plane, the graph of y=f(x) has a minimum value of -16. What is the value of n?
Show the answer and explanation
Why D is right
The vertex x-coordinate is the average of the x-intercepts: x=(2+(−n))/2=(2−n)/2. The minimum value is f((2−n)/2) = ((2−n)/2−2)((2−n)/2+n) = ((−n)/2)((2+n)/2)=−n(2+n)/4=−16. So n(2+n)=64, giving n2 + 2n−64=0. Factoring: (n+8)(n−6)=0, so n=6 or n=−8. Since the parabola opens upward and has a minimum, both values work algebraically, but checking: if n=6, x-intercepts are 2 and -6, vertex at x=−2, f(-2) = -16. If n=−8, x-intercepts are 2 and 8, vertex at x=5, f(5) = -16. The problem expects n=6.
Why the others are wrong
AThis results from setting up n(2 + n)/4 = 16 (missing the negative sign) and solving n² + 2n - 64 = 0, getting n = -8 or 6, then choosing -8.
BThis results from solving n(n + 2) = 64 incorrectly as n² = 16, giving n = ±4.
CThis results from solving n² + 2n - 64 = 0 and choosing the positive root of n² = 16 after misapplying the quadratic formula.
Question 17Hard
The function f is defined by f(x)=3(x−m)2+n, where m and n are constants. In the xy-plane, the vertex of the graph of y=f(x) is (5, -7). What is the value of f(7)?
Show the answer and explanation
Why A is right
The vertex form tells us m=5 and n=−7. Therefore f(x)=3(x−5)2−7. Substituting x=7 gives f(7) = 3(7−5)2−7=3(4) - 7 = 12 - 7 = 5.
Why the others are wrong
BThis incorrectly negates the result, making a sign error in the final calculation.
CThis computes 3(2)² = 12 but forgets to subtract 7.
DThis incorrectly uses the y-coordinate of the vertex as the answer without substituting x = 7.
Question 18Hard
The function f is defined by f(x)=k(x+2)2−3, where k is a constant. In the xy-plane, the graph of y=f(x) passes through the point (-5, 24). For what value of x does f(x) reach its minimum or maximum value?
Show the answer and explanation
Why D is right
The vertex form f(x)=k(x+2)2−3 directly shows the vertex at x=−2 (where the expression (x+2)2 is minimized). The value of k does not change the x-coordinate of the vertex, only whether it is a minimum (k > 0) or maximum (k < 0).
Why the others are wrong
AThis results from confusing the x-coordinate of the given point (-5, 24) with the vertex x-coordinate.
BThis results from a sign error in reading the vertex form, taking h = 2 instead of h = -2 from (x + 2)².
CThis results from confusing the constant term -3 with the x-coordinate of the vertex.
Question 19Hard
A radioactive substance has an initial mass of 200 grams and decays at a rate of 8% per year, compounded quarterly. If M(t) represents the mass in grams after t years, which function models the relationship?
Show the answer and explanation
Why B is right
With 8% annual decay compounded quarterly, the quarterly decay rate is 0.08/4=0.02, so the multiplier is 1−0.02=0.98. Over t years there are 4t quarters, giving M(t)=200(0.98)4t.
Why the others are wrong
AThis uses 0.92 as the quarterly multiplier, incorrectly applying the full 8% decay each quarter instead of dividing by 4.
CThis uses the correct quarterly multiplier 0.98 but counts t as quarters rather than years, missing the 4t exponent.
DThis reverses decay to growth, using 1.02 (a 2% increase) instead of 0.98 (a 2% decrease) each quarter.
Question 20Hard
A chemist models the concentration C, in milligrams per liter, of a medication in a patient's bloodstream using the function C(h)=24(0.78)h, where h is the number of hours after the medication is administered.
What is the best interpretation of the statement "C(8) is approximately equal to 3.5" in this context?
Show the answer and explanation
Why A is right
The function C(h) gives the concentration at time h hours. The statement C(8) ≈ 3.5 means when h=8, the concentration C is approximately 3.5, so the concentration is approximately 3.5 milligrams per liter after 8 hours.
Why the others are wrong
BThis reverses the input and output, incorrectly swapping the time and concentration values.
CThis misinterprets the function value as a rate of change rather than the actual concentration at h = 8.
DThis incorrectly uses 0.78 (the decay factor from the model) as the time value instead of 8.
Question 21Hard
The function f is defined by f(x)=m(x+3)2−12, where m is a constant. If f(1) = 4, what is the minimum value of f(x)?
Show the answer and explanation
Why A is right
From f(1) = 4: m(1+3)2−12=4, so 16m=16 and m=1. Since m > 0, the parabola opens upward and the vertex represents the minimum. The vertex is at (-3, -12), so the minimum value is -12.
Why the others are wrong
BThis results from a sign error in identifying the vertex y-coordinate.
CThis incorrectly assumes the minimum equals the vertex x-coordinate.
DThis results from incorrectly subtracting an additional value when computing the vertex y-coordinate.
Question 22Hard
The function f is defined by f(x)=5(x−r)2+s, where r and s are constants. The graph of y=f(x) in the xy-plane has a vertex at (4, -20) and passes through the point (6, t). What is the value of t?
Show the answer and explanation
Why A is right
The vertex is at (4, -20), so r=4 and s=−20. Thus f(x)=5(x−4)2−20. Substituting x=6: f(6) = 5(2)2−20=20−20=0.
Why the others are wrong
BThis results from incorrectly doubling the vertex y-coordinate.
CThis results from a sign error when computing f(6).
DThis results from incorrectly halving the vertex y-coordinate and using it as t.
Question 23Hard
The function f is defined by f(x)=a(x−3)2+5, where a is a constant. The graph of y=f(x) in the xy-plane passes through the point (1, -3). What is the y-coordinate of the y-intercept of the graph?
Show the answer and explanation
Why B is right
Since the graph passes through (1, -3), we have −3=a(1−3)2+5, which gives −3=4a+5, so a=−2. The y-intercept occurs at x=0: f(0) = -2(0−3)2+5=−2(9) + 5 = -13.
Why the others are wrong
AThis results from incorrectly using a = -4 instead of a = -2.
CThis reflects a sign error in the final calculation.
DThis results from using only the coefficient a without completing the calculation at x = 0.
Question 24Hard
The function f is defined by f(x)=(x−p)(x−q), where p and q are constants. In the xy-plane, the graph of y=f(x) has a minimum value of -16 at x=5. What is the value of pq?
Show the answer and explanation
Why B is right
The vertex of f(x)=(x−p)(x−q) = x2 - (p+q)x + pq is at x=(p+q)/2=5, so p+q=10. The minimum value is f(5)=(5−p)(5−q)=−16. Since 5−p and 5−q are symmetric about 0 when p+q=10, let 5−p=t, then 5−q=−t(because[MATH]p+q=10means(5−p)+(5−q)=0)[/MATH]. So t(−t) = -t2 = -16, giving t2 = 16, thus t=±4. If t=4, then 5−p=4 so p=1, and 5−q=−4 so q=9. Then pq=9. If t=−4, then p=9 and q=1, giving pq=9.
Why the others are wrong
AThis results from confusing the vertex x-coordinate (5) squared with the product pq.
CThis results from correctly finding that (5-p)(5-q) = -16 but incorrectly concluding that pq = -9.
DThis results from confusing the minimum value (-16) with pq.
Question 25Hard
The function f is defined by f(x)=(x−m)(x−n), where m and n are constants with m < n. The minimum value of f(x) is -9, and the graph of y=f(x) in the xy-plane passes through the point (7, 0). What is the value of m?
Show the answer and explanation
Why A is right
Since the graph passes through (7,0), either m=7 or n=7. Given m<n, we have n=7. The vertex x-coordinate is (m+7)/2, and at the vertex f=−9. Expanding: f(x) = x2 - (m+7)x+7m. The minimum occurs at x=(m+7)/2, where f = -((m+7)/2−m)((m+7)/2−7) = -((7−m)/2)((m−7)/2) = -(7−m)2/4=−9. So (7−m)2=36, giving 7−m=6 (taking positive root since m<7), thus m=1.
Why the others are wrong
BThis applies an incorrect negative sign when solving the vertex equation for m.
CThis uses the x-coordinate of the vertex (m + 7)/2 as the value of m directly without solving the minimum condition.
DThis incorrectly identifies m with the given x-intercept rather than recognizing n = 7 and solving for m.
These 25 are a sample, not a study plan
These 25 come from a bank of 17,599 questions across all 31 SAT skills and three difficulty tiers. Inside SAT Climb you get the rest of Nonlinear functions, a diagnostic that finds which skills are actually costing you points, and a grid that shows what to drill next.
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