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.
Off-by-one from a hidden “after the first …” offset.
Swaps slope and intercept when reading the prose.
Extrapolates past the given points without using the rate.
Question 1Hard
The function f is defined by f(x) = kx - 8, where k is a constant. If f(2) = 6, what is the value of f(-3)?
Show the answer and explanation
Why A is right
Since f(2) = 6 and f(x) = kx - 8, substituting gives 6=2k−8, so 2k=14 and k=7. Therefore f(x)=7x−8, and f(-3) = 7(-3) - 8 = -21 - 8 = -29.
Why the others are wrong
BThis results from using k = -3 instead of k = 7, or from computing 7(-3) = -21 but then adding 8 instead of subtracting 8, yielding -21 + 8 = -13, or another coefficient error.
CThis results from a sign error, such as computing f(-3) = 7(3) - 8 = 13 by dropping the negative sign on the input.
DThis results from swapping the operation, computing k - 3 instead of k(-3), yielding 7 - 3 - 8 = -4, or from computing -3 - 8 = -11 without applying the coefficient.
Question 2Hard
A savings account contains $1200. The account holder deposits $85 each week. Which function gives the total amount A, in dollars, in the account after w weeks?
Show the answer and explanation
Why D is right
The account starts with $1200 and grows by $85 per week. After w weeks, the total deposited is 85w dollars, so the total amount is A(w)=1200+85w, which equals 85w+1200.
Why the others are wrong
AThis incorrectly adds the initial amount and weekly deposit together as a single coefficient, eliminating the constant term.
BThis incorrectly subtracts the weekly deposit instead of adding it, which would represent withdrawals rather than deposits.
CThis incorrectly treats the initial amount as the rate per week and the weekly deposit as the initial amount.
Question 3Hard
A water tank's volume V, in gallons, is modeled by V(h)=12.5h+150, where h is the number of hours after a valve is opened.
What is the best interpretation of 150 in this context?
Show the answer and explanation
Why B is right
The y-intercept of a linear function represents the initial value when the independent variable is zero. When h=0, V(0) = 150, meaning the tank initially contains 150 gallons.
Why the others are wrong
AThis confuses the intercept with the slope and uses the wrong time unit; the rate of change is 12.5 gallons per hour, not per minute.
CThis confuses the intercept (150) with the slope (12.5), which represents the rate of change.
DThis misinterprets 150 as a time value rather than the initial volume.
Question 4Hard
The function f is defined by f(x) = ax + b, where a and b are constants. If f(2) = 11 and f(6) = 27, what is the value of f(−1)?
Show the answer and explanation
Why A is right
From f(2) = 11 and f(6) = 27, we have 2a+b=11 and 6a+b=27. Subtracting the first from the second gives 4a=16, so a=4. Substituting back, 2(4)+b=11 gives b=3. Therefore f(x)=4x+3, and f(−1) = 4(−1) + 3 = −4 + 3 = −1.
Why the others are wrong
BThis is the result of a sign error, computing f(−1) = 4(−1) + 3 as 4 + 3 − 4 = 1, incorrectly handling the negative input.
CThis is the result of using b = 3 directly without computing 4(−1), essentially evaluating f(−1) as just b or computing −1 + 4 = 3, an off-by-coefficient error.
DThis is the result of subtracting instead of adding, computing f(−1) = b − a = 3 − (−4) = 7, a swapped operation in the function evaluation.
Question 5Hard
The function m is defined by m(x)=6x−30. What is the value of the product of the x-intercept and the y-intercept of the graph of y=m(x) in the xy-plane?
Show the answer and explanation
Why D is right
The y-intercept occurs when x=0, so m(0) = 6(0) - 30 = -30. The x-intercept occurs when m(x)=0, so 6x−30=0, giving x=5. The product of the x-intercept and y-intercept is 5 × (-30) = -150.
Why the others are wrong
AThis results from computing 5 × 5 = 25, squaring the x-intercept instead of multiplying it by the y-intercept.
BThis results from correctly computing the product but dropping the negative sign, giving 150 instead of -150.
CThis results from adding the intercepts with a sign error, computing 5 + (-30) = -25 instead of multiplying them.
Question 6Hard
The function g is defined by g(x) = ax + b. If g(4) = 15 and g(10) = 27, what is the value of g(20)?
Show the answer and explanation
Why D is right
Using g(4) = 15 and g(10) = 27, we have 4a+b=15 and 10a+b=27. Subtracting the first from the second gives 6a=12, so a=2. Substituting back yields 4(2)+b=15, so b=7. Therefore g(x)=2x+7, and g(20) = 2(20) + 7 = 40 + 7 = 47.
Why the others are wrong
AThis results from using a = 1 instead of a = 2, computing 1(20) + 0 = 20 without correctly solving for both parameters.
BThis results from using a = -2 instead of a = 2, a sign error, then computing -2(20) + 33 = -7.
CThis results from computing a(20 - 10) + 15 instead of a(20) + b, swapping how the interval relationship applies.
Question 7Hard
The function f is defined by f(x) = cx + 9, where c is a constant. If f(-3) = 27, what is the value of f(4)?
Show the answer and explanation
Why A is right
Since f(-3) = 27 and f(x) = cx + 9, substituting gives 27=c(-3) + 9, so −3c=18 and c=−6. Therefore f(x)=−6x+9, and f(4) = -6(4) + 9 = -24 + 9 = -15.
Why the others are wrong
BThis results from using an incorrect coefficient, such as c = -7, yielding f(4) = -7(4) + 9 = -28 + 9 = -19, or from other coefficient errors leading to -21.
CThis results from a sign error in solving for c, obtaining c = 6 instead of c = -6, yielding f(4) = 6(4) + 9 = 24 + 9 = 33, or from dropping the negative sign in the final calculation.
DThis results from swapping operations, such as computing c + 4 instead of c(4), yielding -6 + 4 + 9 = 7, or from computing 4 + 9 = 13 without applying the coefficient.
Question 8Hard
The function h is defined by h(x) = px + q, where p and q are constants. If h(4) = 17 and h(7) = 29, what is the value of h(10)?
Show the answer and explanation
Why B is right
From h(4) = 17 and h(7) = 29, we have 4p+q=17 and 7p+q=29. Subtracting the first from the second gives 3p=12, so p=4. Substituting back, 4(4)+q=17 gives q=1. Therefore h(x)=4x+1, and h(10) = 4(10) + 1 = 41.
Why the others are wrong
AThis results from using p = 3 instead of p = 4, then computing 3(10) + 7 = 37, misidentifying the slope.
CThis results from computing h(10) = 10 + 35, treating the function as additive rather than using correct multiplication.
DThis results from sign errors in solving the system, obtaining incorrect p or q values and computing a negative result.
Question 9Hard
The function k is defined by k(x)=−5x−15. What is the value of the sum of the x-intercept and the y-intercept of the graph of y=k(x) in the xy-plane?
Show the answer and explanation
Why C is right
The y-intercept occurs when x=0, so k(0) = -5(0) - 15 = -15. The x-intercept occurs when k(x)=0, so −5x−15=0, which gives x=−3. The sum of the x-intercept and y-intercept is (-3) + (-15) = -18.
Why the others are wrong
AThis results from computing (-3) - (-15) = 12 or -3 + (-15) = -18 but then making an arithmetic slip to get -12.
BThis is the product of the x-intercept and y-intercept ((-3) × (-15) = 45) instead of their sum.
DThis results from a sign error, computing the absolute value |(-3) + (-15)| = 18 or computing 3 + 15 = 18.
Question 10Hard
The function h is defined by h(x) = px + 12. If h(5)=h(9) + 24, what is the value of h(-3)?
Show the answer and explanation
Why C is right
Since h(x) = px + 12, we have h(5)=5p+12 and h(9)=9p+12. The condition h(5)=h(9) + 24 becomes 5p+12=9p+12+24, so 5p=9p+24, yielding −4p=24 and p=−6. Therefore h(x)=−6x+12, and h(-3) = -6(-3) + 12 = 18 + 12 = 30.
Why the others are wrong
AThis results from computing p(-3 + 12) instead of p(-3) + 12, swapping the order of operations.
BThis results from using p = -2 instead of correctly solving for p = -6, then computing -2(-3) + 12 = -18.
DThis results from using p = 6 instead of p = -6, a sign error when solving -4p = 24.
Question 11Hard
The function f is defined by f(x) = kx − 8, where k is a constant. If f(4) = 20, what is the value of f(−3)?
Show the answer and explanation
Why A is right
Since f(x) = kx − 8 and f(4) = 20, substituting gives 20=4k − 8. Solving yields 28=4k, so k=7. Therefore f(x)=7x − 8, and f(−3) = 7(−3) − 8 = −21 − 8 = −29.
Why the others are wrong
BThis is the result of adding k and the input instead of multiplying, computing f(−3) = 7 + (−3) − 8 = −4, or adding instead of subtracting the constant, computing −21 + 8 = −13.
CThis is the result of a sign error, computing f(−3) = 7(3) − 8 = 21 − 8 = 13, treating the input as positive instead of negative.
DThis is the result of using k = −3 or the input value directly, computing f(−3) = (−3) − 8 + 6 = −5, an off-by-coefficient error where the slope is not properly applied.
Question 12Hard
The function f is defined by f(x) = hx + n, where h and n are constants. If f(2) = 11 and f(5) = 20, what is the value of f(-1)?
Show the answer and explanation
Why B is right
From f(2) = 11 and f(5) = 20, we have 2h+n=11 and 5h+n=20. Subtracting the first from the second gives 3h=9, so h=3. Substituting back gives n=5. Therefore f(x)=3x+5, and f(-1) = 3(-1) + 5 = -3 + 5 = 2.
Why the others are wrong
AThis results from adding -1 to 5 or using an incorrect coefficient, treating the operation as simple addition.
CThis results from using h = -3 instead of h = 3, yielding f(-1) = -3(-1) + 5 = 3 + 5 = 8, or a sign error elsewhere.
DThis results from using h = 2 instead of h = 3, yielding f(-1) = 2(-1) + 5 = -2 + 5 = 3, or another coefficient error.
Question 13Hard
The function h is defined by h(x)=4x+20. What is the value of the product of the x-intercept and the y-intercept of the graph of y=h(x) in the xy-plane?
Show the answer and explanation
Why A is right
The y-intercept occurs when x=0, so h(0) = 4(0) + 20 = 20. The x-intercept occurs when h(x)=0, so 4x+20=0, which gives x=−5. The product of the x-intercept and y-intercept is (-5) × 20 = -100.
Why the others are wrong
BThis results from taking the absolute value of the product or making a sign error, computing 5 × 20 = 100.
CThis is the sum of the x-intercept and y-intercept ((-5) + 20 = 15) instead of their product.
DThis results from computing (-5) × 5 = -25, incorrectly identifying the y-intercept as 5 instead of 20.
Question 14Hard
The function f is defined by f(x) = wx - 14, where w is a constant. If f(11) = 52, what is the value of f(16)?
Show the answer and explanation
Why C is right
Since f(11) = 52 and f(x) = wx - 14, substituting gives 52=11w−14, so 11w=66 and w=6. Therefore f(x)=6x−14, and f(16) = 6(16) - 14 = 96 - 14 = 82.
Why the others are wrong
AThis results from subtracting 14 from 16 instead of computing f(16) after finding w = 6.
BThis results from a sign error in the final computation, such as computing -(6(16) - 14) instead of 6(16) - 14.
DThis results from using w = 16 as the coefficient instead of solving for w = 6 from the given condition.
Question 15Hard
The function f is defined by f(x) = px - 15, where p is a constant. If f(5) = 35, what is the value of f(-2)?
Show the answer and explanation
Why A is right
Since f(5) = 35 and f(x) = px - 15, substituting gives 35=5p−15, so 5p=50 and p=10. Therefore f(x)=10x−15, and f(-2) = 10(-2) - 15 = -20 - 15 = -35.
Why the others are wrong
BThis results from using an incorrect value of p, such as p = 5, yielding f(-2) = 5(-2) - 15 = -10 - 15 = -25, or from computing 10(-2) = -20 but only subtracting part of the constant.
CThis results from a sign error, such as computing f(-2) = 10(2) - 15 = 5 by dropping the negative sign on the input.
DThis results from swapping operations, computing p - 2 instead of p(-2), yielding 10 - 2 - 15 = -7, or from computing -2 - 15 = -17 without applying the coefficient.
Question 16Hard
The function h is defined by h(x) = rx + s, where r and s are constants. If h(3) = 14 and h(8) = 34, what is the value of h(12)?
Show the answer and explanation
Why B is right
From h(3) = 14 and h(8) = 34, we have 3r+s=14 and 8r+s=34. Subtracting gives 5r=20, so r=4. Substituting back, 3(4)+s=14 gives s=2. Therefore h(x)=4x+2, and h(12) = 4(12) + 2 = 50.
Why the others are wrong
AThis results from computing h(12) = 12 + 34, treating the function as additive rather than using correct multiplication.
CThis results from using r = 5 instead of r = 4, then computing 5(12) - 6 = 54, misidentifying the coefficient.
DThis results from sign errors in solving the system, obtaining r = -4 and computing -4(12) + 46 = -2 or similar error.
Question 17Hard
The function t is defined by t(x)=−7x−28. What is the value of the sum of the x-intercept and the y-intercept of the graph of y=t(x) in the xy-plane?
Show the answer and explanation
Why B is right
The y-intercept occurs when x=0, so t(0) = -7(0) - 28 = -28. The x-intercept occurs when t(x)=0, so −7x−28=0, which gives x=−4. The sum of the x-intercept and y-intercept is (-4) + (-28) = -32.
Why the others are wrong
AThis is the product of the x-intercept and y-intercept ((-4) × (-28) = 112) instead of their sum.
CThis results from computing (-4) - (-28) = 24 or another arithmetic slip that yields -24.
DThis results from a sign error, computing the absolute value |(−4) + (−28)| = 32 or computing 4 + 28 = 32.
Question 18Hard
The function f is defined by f(x) = cx + 22, where c is a constant. If f(8) = 6, what is the value of f(12)?
Show the answer and explanation
Why C is right
Since f(8) = 6 and f(x) = cx + 22, substituting gives 6=8c+22, so 8c=−16 and c=−2. Therefore f(x)=−2x+22, and f(12) = -2(12) + 22 = -24 + 22 = -2.
Why the others are wrong
AThis results from subtracting 8 from 22 instead of computing f(12) after finding c.
BThis results from a sign error when finding c, such as computing c = 2 instead of c = -2.
DThis results from using c = 12 as the coefficient without solving for c from the given condition.
Question 19Hard
The function f is defined by f(x) = tx - 15. If f(3) = 9, what is the value of f(-5)?
Show the answer and explanation
Why B is right
Since f(x) = tx - 15 and f(3) = 9, substituting gives 3t−15=9, so 3t=24 and t=8. Therefore f(x)=8x−15, and f(-5) = 8(-5) - 15 = -40 - 15 = -55.
Why the others are wrong
AThis results from a sign error in computing f(-5), incorrectly treating -5 as positive 5, giving f(5) = 8(5) - 15 = 40 - 15 = 25.
CThis results from using t = -5 directly as the coefficient without solving for t from the given condition, then computing -5 - 15 = -20 or making a coefficient error.
DThis results from treating the operation as addition instead of multiplication and subtraction, computing -5 + (-15)/something or making a similar operation swap.
Question 20Hard
The function g is defined by g(x) = cx + d, where c and d are constants. If g(2) = 11 and g(5) = 20, what is the value of g(8)?
Show the answer and explanation
Why C is right
From g(2) = 11 and g(5) = 20, we have 2c+d=11 and 5c+d=20. Subtracting gives 3c=9, so c=3. Substituting back, 2(3)+d=11 gives d=5. Therefore g(x)=3x+5, and g(8) = 3(8) + 5 = 29.
Why the others are wrong
AThis results from using c = 2 instead of c = 3, then computing 2(8) + 7 = 23, misidentifying the coefficient.
BThis results from computing g(8) = 8 + 18, treating the function as additive components rather than using correct multiplication.
DThis results from sign errors in solving the system, obtaining c = -3 or negative d and computing a negative result.
Question 21Hard
The function g is defined by g(x)=−2x+10. What is the value of the sum of the x-intercept and the y-intercept of the graph of y=g(x) in the xy-plane?
Show the answer and explanation
Why C is right
The y-intercept occurs when x=0, so g(0) = -2(0) + 10 = 10. The x-intercept occurs when g(x)=0, so −2x+10=0, which gives x=5. The sum of the x-intercept and y-intercept is 5 + 10 = 15.
Why the others are wrong
AThis is the product of the x-intercept and y-intercept (5 × 10 = 50) instead of their sum.
BThis results from an arithmetic error, possibly computing 10 - 2 = 8 or misidentifying one of the intercepts.
DThis results from computing 5 - 10 = -5, using subtraction in the wrong order instead of addition.
Question 22Hard
The function f is defined by f(x) = ax + 15, where a is a constant. If f(6) = 3, what is the value of f(9)?
Show the answer and explanation
Why D is right
Since f(6) = 3 and f(x) = ax + 15, substituting gives 3=6a+15, so 6a=−12 and a=−2. Therefore f(x)=−2x+15, and f(9) = -2(9) + 15 = -18 + 15 = -3.
Why the others are wrong
AThis results from subtracting 6 from 15 instead of computing f(9) after finding a = -2.
BThis results from a sign error when finding a, such as computing a = 2 instead of a = -2.
CThis results from using a = 9 as the coefficient without solving for a from the given condition.
Question 23Hard
In the xy-plane, a line passes through the points (-5, 22) and (3, -10). If the function f defines this line, what is the value of f(7)?
Show the answer and explanation
Why A is right
The slope of the line is (-10 - 22)/(3 - (-5)) = -32/8 = -4. Using point-slope form with (3, -10): f(x)−(−10)=−4(x−3), so f(x)=−4x+12−10=−4x+2. Therefore f(7) = -4(7) + 2 = -28 + 2 = -26.
Why the others are wrong
BThis results from an error in the y-intercept, such as f(x) = -4x + 10 instead of -4x + 2, yielding f(7) = -28 + 10 = -18.
CThis results from a sign error in the slope, using f(x) = 4x + 2 instead of -4x + 2, yielding f(7) = 28 + 2 = 30, or from other sign errors leading to 26.
DThis results from swapping operations or using f(7) = f(3) without properly computing, or from computing -4 + 7 + 2 = 5 through operation confusion, or from using the y-value at x = 3 directly.
Question 24Hard
The function g is defined by g(x) = mx + n, where m and n are constants. If g(1) = 8 and g(6) = 33, what is the value of g(11)?
Show the answer and explanation
Why B is right
From g(1) = 8 and g(6) = 33, we have m+n=8 and 6m+n=33. Subtracting gives 5m=25, so m=5. Substituting back, 5+n=8 gives n=3. Therefore g(x)=5x+3, and g(11) = 5(11) + 3 = 58.
Why the others are wrong
AThis results from using m = 4 instead of m = 5, then computing 4(11) + 9 = 53, misidentifying the coefficient.
CThis results from computing g(11) = 11 + 33 = 44, treating the function as additive rather than using correct multiplication.
DThis results from sign errors in solving the system, obtaining m = -5 and computing -5(11) + 52 = -3 or similar error.
Question 25Hard
The function f is defined by f(x) = wx + 18. If f(4) = 6, what is the value of f(-6)?
Show the answer and explanation
Why C is right
Since f(x) = wx + 18 and f(4) = 6, substituting gives 4w+18=6, so 4w=−12 and w=−3. Therefore f(x)=−3x+18, and f(-6) = -3(-6) + 18 = 18 + 18 = 36.
Why the others are wrong
AThis results from using w = -6 directly as the coefficient without solving for w from the given condition, then computing -6 + 18 = 12 or making a coefficient error.
BThis results from a sign error in computing f(-6), incorrectly treating -6 as positive 6, giving f(6) = -3(6) + 18 = -18 + 18 = 0, or making another sign error.
DThis results from treating the operation incorrectly, subtracting instead of multiplying and adding, or computing -6 + 6 = 0.
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 Linear functions, a diagnostic that finds which skills are actually costing you points, and a grid that shows what to drill next.
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.