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.
The function f is defined by f(x)=7x−12. What is the value of f(x−3)?
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Why A is right
To find f(x−3), substitute (x−3) for x in the expression f(x)=7x−12. This gives f(x−3)=7(x−3)−12=7x−21−12=7x−33.
Why the others are wrong
BThis results from subtracting 3 from the constant term only, rather than substituting (x - 3) into the entire function.
CThis results from a sign error when combining -21 and -12, yielding +9 instead of -33.
DThis results from using the coefficient 1 instead of 7 when multiplying (x - 3).
Question 2Medium
A savings account has a balance of 500 dollars. After the first month, 40 dollars is deposited into the account each month. Which function gives the balance b, in dollars, in the account after m months?
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Why D is right
The account starts with 500 dollars. Deposits begin after the first month, so after m months, deposits have been made for (m−1) months. The function is b=500+40(m−1).
Why the others are wrong
AThis reverses the initial balance and monthly deposit rate, using 40 as the base and 500 as the rate.
BThis incorrectly subtracts 40 from the initial balance instead of properly accounting for the deposit offset.
CThis incorrectly counts the first month as a month when a deposit was made, ignoring the 'after the first month' condition.
Question 3Medium
A cell phone plan charges a monthly fee of 35 dollars plus 0.12 dollars per text message sent. The function f(n)=35+0.12n gives the total monthly cost, in dollars, for sending n text messages. What is the best interpretation of 35 in this context?
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Why C is right
The function f(n)=35+0.12n is in slope-intercept form where 35 is the y-intercept, representing the cost when n=0. This is the fixed monthly fee that is charged regardless of how many text messages are sent.
Why the others are wrong
AThis describes the slope 0.12, not the y-intercept 35.
BThis reverses the interpretation; 35 is not related to a quantity of messages but to the base cost.
DThis incorrectly introduces subtraction when the function shows addition, and misidentifies what happens at n = 0.
Question 4Medium
The height of water in a tank is modeled as a linear function of the number of gallons added. The height was 14 inches when 20 gallons had been added and 26 inches when 44 gallons had been added. Based on the model, what is the height, in inches, when 32 gallons have been added?
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Why C is right
The slope is (26 - 14)/(44 - 20) = 12/24 = 0.5 inches per gallon. Using point-slope form with the point (20, 14): h=14+0.5(g−20). At g=32 gallons: h=14+0.5(32 - 20) = 14 + 0.5(12) = 14 + 6 = 20 inches.
Why the others are wrong
AThis is the slope of the linear function (rate of change per gallon), not the height at 32 gallons.
BThis results from an arithmetic error when computing 14 + 0.5(12), such as calculating 14 + 4 = 18 instead of 14 + 6 = 20.
DThis results from incorrectly averaging the two given heights (14 + 26)/2 = 20 and then adding 3, or from computing the change without proper evaluation of the linear model.
Question 5Medium
The function g is defined by g(x)=−2x+10. What is the value of the sum of the x-intercept and y-intercept of the graph of y=g(x) in the xy-plane?
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Why B is right
The x-intercept occurs when g(x)=0, so −2x+10=0, giving x=5. The y-intercept occurs when x=0, so g(0) = 10. The sum is 5 + 10 = 15.
Why the others are wrong
AThis results from an arithmetic error in solving for the x-intercept, possibly finding x = -2 instead of x = 5.
CThis is the product of the x-intercept and y-intercept (5 × 10 = 50) rather than their sum.
DThis results from subtracting instead of adding the intercepts (10 - 5 = 5, with possible further error) or other arithmetic slip.
Question 6Medium
A container holds 80 liters of solution. After the first hour, solution is drained from the container at a constant rate of 6 liters per hour. Which function gives the amount of solution s, in liters, remaining in the container after h hours?
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Why D is right
The container starts with 80 liters. Draining begins after the first hour, so after h hours, solution has been drained for (h−1) hours. The function is s=80−6(h−1).
Why the others are wrong
AThis incorrectly subtracts 6 from the initial amount and uses addition instead of subtraction for the draining process.
BThis reverses the initial amount and drain rate, and uses incorrect signs.
CThis incorrectly counts the first hour as an hour when solution was drained, ignoring the 'after the first hour' condition.
Question 7Medium
A spring has a natural length of 22 centimeters. When a weight is attached, the spring stretches at a constant rate of 3 centimeters per kilogram of weight added. The function f(w)=22+3w gives the length of the spring, in centimeters, when a weight of w kilograms is attached. What is the best interpretation of 22 in this context?
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Why B is right
The function f(w)=22+3w is in slope-intercept form where 22 is the y-intercept, representing the length when w=0. This is the natural length of the spring before any weight is attached.
Why the others are wrong
AThis describes the slope 3, not the y-intercept 22.
CThis misinterprets the constant as a limit on weight capacity, when it actually represents the initial length.
DThis introduces compression and removal when the context describes stretching and addition, and misidentifies the y-intercept.
Question 8Medium
The temperature of a chemical reaction is modeled as a linear function of time. The temperature was 18 degrees Celsius at 5 minutes and 42 degrees Celsius at 13 minutes. Based on the model, what is the temperature, in degrees Celsius, at 9 minutes?
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Why B is right
The slope is (42 - 18)/(13 - 5) = 24/8 = 3 degrees per minute. Using point-slope form with the point (5, 18): T=18+3(t−5). At t=9 minutes: T=18+3(9 - 5) = 18 + 3(4) = 18 + 12 = 30 degrees Celsius.
Why the others are wrong
AThis results from an arithmetic error when computing 18 + 3(4), such as calculating 18 + 9 = 27 instead of 18 + 12 = 30.
CThis is the slope of the linear function (rate of change), not the temperature at 9 minutes.
DThis results from using addition instead of the linear model evaluation, such as computing 18 + 18 = 36 or incorrectly finding the midpoint.
Question 9Medium
The function k is defined by k(x)=−5x−15. What is the value of the sum of the x-intercept and y-intercept of the graph of y=k(x) in the xy-plane?
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Why C is right
The x-intercept occurs when k(x)=0, so −5x−15=0, giving x=−3. The y-intercept occurs when x=0, so k(0) = -15. The sum is (-3) + (-15) = -18.
Why the others are wrong
AThis is the product of the x-intercept and y-intercept ((-3) × (-15) = 45) rather than their sum.
BThis results from an arithmetic error, possibly computing -15 - (-3) = -12 or similar sign mistake.
DThis results from taking the absolute value of the sum or making a sign error in adding the intercepts.
Question 10Medium
A tree is 8 feet tall. After the first week, the tree grows at a constant rate of 0.5 feet per week. Which function gives the height h, in feet, of the tree after w weeks?
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Why C is right
The tree starts at 8 feet. Growth begins after the first week, so after w weeks, the tree has grown for (w−1) weeks. The function is h=8+0.5(w−1).
Why the others are wrong
AThis incorrectly includes the first week as a week of growth, ignoring the 'after the first week' condition.
BThis incorrectly subtracts 0.5 from the initial height instead of properly accounting for the growth offset.
DThis reverses the initial height and growth rate, using 0.5 as the base and 8 as the rate.
Question 11Medium
A candle is 18 centimeters tall when lit. The candle burns at a constant rate, decreasing in height by 2.5 centimeters per hour. The function f(h)=18−2.5h gives the height of the candle, in centimeters, after burning for h hours. What is the best interpretation of 2.5 in this context?
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Why D is right
The function f(h)=18−2.5h shows that for each hour h, the height decreases by 2.5 centimeters. The coefficient 2.5 represents the rate of change in height per hour, which is a decrease since it is subtracted.
Why the others are wrong
AThis describes the y-intercept 18, not the slope coefficient 2.5.
BThis misinterprets the sign; since 2.5h is subtracted, the height decreases rather than increases.
CThis confuses the burn rate with total time. The time to burn completely would require dividing 18 by 2.5.
Question 12Medium
The function p is defined by p(x)=−4x+24. What is the value of the sum of the x-intercept and y-intercept of the graph of y=p(x) in the xy-plane?
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Why C is right
The x-intercept occurs when p(x)=0, so −4x+24=0, giving x=6. The y-intercept occurs when x=0, so p(0) = 24. The sum is 6 + 24 = 30.
Why the others are wrong
AThis is the product of the x-intercept and y-intercept (6 × 24 = 144) rather than their sum.
BThis results from subtracting instead of adding (24 - 6 = 18) or another arithmetic slip.
DThis results from an error in computing the x-intercept, possibly finding x = -6 and then computing -6 + (-12) or similar mistake.
Question 13Medium
A bathtub contains 12 gallons of water. Water is being added to the bathtub at a constant rate of 4.5 gallons per minute. The function f(t)=12+4.5t gives the amount of water, in gallons, in the bathtub t minutes after water begins to be added. What is the best interpretation of 4.5 in this context?
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Why D is right
The function f(t)=12+4.5t shows that for each minute t, 4.5 gallons are added to the water already in the bathtub. The coefficient 4.5 represents the rate at which water is being added.
Why the others are wrong
AThis misinterprets the sign; since 4.5t is added, water is being added to the bathtub, not drained.
BThis describes the y-intercept 12, not the slope coefficient 4.5.
CThis confuses the fill rate with total time. The time to reach a certain volume would depend on both the initial amount and the rate.
Question 14Medium
The function m is defined by m(x)=6x−18. What is the value of the product of the x-intercept and y-intercept of the graph of y=m(x) in the xy-plane?
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Why D is right
The x-intercept occurs when m(x)=0, so 6x−18=0, giving x=3. The y-intercept occurs when x=0, so m(0) = -18. The product is 3 × (-18) = -54.
Why the others are wrong
AThis is the sum of the x-intercept and y-intercept (3 + (-18) = -15) rather than their product.
BThis results from correctly computing the product but dropping the negative sign.
CThis results from adding the absolute values (3 + 18 = 21) instead of multiplying the intercepts.
Question 15Medium
A truck rental company charges a flat fee of 60 dollars plus 0.85 dollars per mile driven. The function f(m)=60+0.85m gives the total rental cost, in dollars, for driving m miles. What is the best interpretation of 0.85 in this context?
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Why A is right
The function f(m)=60+0.85m shows that for each mile m driven, an additional 0.85 dollars is added to the cost. The coefficient 0.85 represents the per-mile charge or cost per mile driven.
Why the others are wrong
BThis describes the y-intercept 60, not the slope coefficient 0.85.
CThis reverses the relationship; 0.85 is the cost per mile, not the mileage obtained for 0.85 dollars.
DThis misinterprets the operation; since 0.85m is added, it represents an additional charge per mile, not a discount.
Question 16Medium
The function h is defined by h(x)=4x+20. What is the value of the product of the x-intercept and y-intercept of the graph of y=h(x) in the xy-plane?
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Why A is right
The x-intercept occurs when h(x)=0, so 4x+20=0, giving x=−5. The y-intercept occurs when x=0, so h(0) = 20. The product is (-5) × 20 = -100.
Why the others are wrong
BThis results from correctly computing the product but dropping the negative sign.
CThis is the sum of the x-intercept and y-intercept (-5 + 20 = 15) rather than their product.
DThis results from adding the absolute values (5 + 20 = 25) instead of multiplying the intercepts.
Question 17Medium
A coffee shop has 240 pounds of coffee beans in stock. The shop uses coffee beans at a constant rate of 18 pounds per day. The function f(d)=240−18d gives the amount of coffee beans, in pounds, remaining after d days. What is the best interpretation of 240 in this context?
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Why C is right
The function f(d)=240−18d has 240 as the y-intercept, representing the amount of coffee beans when d=0. This is the initial stock of coffee beans before any are used.
Why the others are wrong
AThis confuses the initial quantity with the time duration. The number of days would require dividing 240 by 18.
BThis describes the slope 18, not the y-intercept 240.
DThis misinterprets the sign; since 18d is subtracted, beans are being used (depleted), not restocked.
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.
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.