Digital SAT Math Practice Problems: Exponential Functions
Exponential functions questions on Digital SAT Math cluster heavily around real-world growth and decay scenarios: population growth, radioactive decay, bacterial cultures, compound interest, and depreciation. The exam tests whether you can translate a percent increase or decrease into the correct base (1 + r for growth, 1 − r for decay), recognize the difference between a constant ratio (exponential) and a constant difference (linear) in a table, and correctly interpret what a specific function value or the y-intercept means in context.
The 18 problems below cover this full range, including growth and decay word problems, table-reading questions that test the exponential-vs-linear distinction, graph identification, and several "best interpretation" questions that reward reading carefully rather than just computing. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. For the interpretation questions in particular, pay close attention to which quantity is the input and which is the output — that's where most wrong-answer traps live. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
1Identifying a Function from Its Shifted Graph
The graph of y = f(x) − 1 is shown. Which equation could define function f?
Graph of y = f(x) − 1
A) f(x) = 2^x
B) f(x) = 2^x − 1
C) f(x) = 2^x + 1
D) f(x) = 2^x + 2
Answer Explanation
The graph appears to pass through (0, 2) and reach about (2, 5). Since the graph shows y = f(x) − 1, adding 1 back gives f(x)'s actual values: f(0) should be 3, and f(2) should be 6.
Testing option D, f(x) = 2^x + 2: f(0) = 1 + 2 = 3 ✓, and f(2) = 4 + 2 = 6 ✓. Subtracting 1 from this function gives 2^x + 1, which passes through (0, 2) and (2, 5), matching the graph. The correct answer is D.
2Interpreting a Constant in a Model
An acceptable noise criterion rating for the background noise in a laundry room is 50. For a noise criterion rating of 50, the equation y = 22(0.997)^(x−60) + 47 gives the estimated sound pressure level, y, in decibels, as a function of the octave band center frequency, x, in hertz, where x ≥ 60. Which of the following is the best interpretation of 47 in this context?
A) 47 is 22 less than the estimated sound pressure level, in decibels, at an octave band center frequency of 60 hertz.
B) 47 is 22 less than the estimated sound pressure level, in decibels, at an octave band center frequency of 0 hertz.
C) 47 is the estimated sound pressure level, in decibels, at an octave band center frequency of 60 hertz.
D) 47 is the estimated sound pressure level, in decibels, at an octave band center frequency of 0 hertz.
Answer Explanation
Evaluate the function at x = 60: y = 22(0.997)⁰ + 47 = 22(1) + 47 = 69. So the sound pressure level at 60 hertz is 69 decibels — not 47 itself.
Since 69 − 22 = 47, the constant 47 represents a value 22 less than the sound pressure level at 60 hertz (the boundary of the domain, not 0 hertz, since x ≥ 60). The correct answer is A.
3Writing an Exponential Growth Model
By examining pollen in the soil, scientists estimated that the number of Ulmus trees in an ancient population doubled every 664 years. There were n trees in the earliest known sample, where n is a constant. Which expression gives the estimated number of Ulmus trees x years after the year of the earliest known sample?
A) n(2)^(664x)
B) n(2)^(664/x)
C) n(2)^(x/664)
D) n(2)^(664+x)
Answer Explanation
A doubling time of 664 years means the population doubles every time x increases by 664 — that's captured by putting x/664 in the exponent, so that when x = 664, the exponent equals exactly 1 (one full doubling).
That gives n(2)^(x/664). The correct answer is C.
4Finding a Y-Intercept
f(x) = 5^x + 3. If the given function f is graphed in the xy-plane, where y = f(x), what is the y-intercept of the graph?
A) (0, 3)
B) (0, 4)
C) (0, 5)
D) (0, 8)
Answer Explanation
The y-intercept occurs at x = 0: f(0) = 5⁰ + 3 = 1 + 3 = 4.
The y-intercept is (0, 4). The correct answer is B.
5Identifying a Decreasing Exponential Table
Which table could represent values of x and their corresponding values of f(x) for a decreasing exponential function f?
A)
x
f(x)
1
1
2
2
3
3
4
4
B)
x
f(x)
1
8
2
4
3
2
4
1
C)
x
f(x)
1
1
2
9
4
17
8
25
D)
x
f(x)
1
25
2
17
4
9
8
1
Answer Explanation
An exponential function has a constant ratio between consecutive outputs (not a constant difference, which would signal a linear function).
Table B has values 8, 4, 2, 1 — each term is exactly half the one before it (a constant ratio of 0.5), and the values are decreasing. The other tables either show a constant difference (linear, not exponential) or aren't consistently decreasing. The correct answer is B.
6Writing an Exponential Growth Model
A certain neighborhood had a population of 1,340 in 2009. Each year for the next 5 years, the population of the neighborhood increased by approximately 3% of the population the previous year. Which of the following equations represents the population, N, of the neighborhood t years after 2009, where t ≤ 5?
A) N = 0.03(1,340)^t
B) N = 1,340(0.03)^t
C) N = 1,340(1.03)^t
D) N = 1.03(1,340)^t
Answer Explanation
A 3% annual increase means each year's population is 103% of the previous year's, giving a growth factor of 1.03. The starting population, 1,340, is the initial value (coefficient).
That gives N = 1,340(1.03)^t. The correct answer is C.
7Solving for Constants from Two Points
The function f is defined by f(x) = a^x − b, where a and b are constants. The graph of y = f(x) in the xy-plane passes through the points (0, −5) and (1, 2). What is the value of a?
A) 8
B) 7
C) 6
D) 5
Answer Explanation
Use (0, −5): a⁰ − b = 1 − b = −5, so b = 6.
Use (1, 2): a¹ − b = a − 6 = 2, so a = 8. The correct answer is A.
8Interpreting a Function Value in Context
f(x) = 6.4(0.5)^(3x). A sample of carbon-11 radioactively decays. The function f models the mass, in grams (g), of carbon-11 in the sample x hours after it is initially measured. Which statement is the best interpretation of f(2) = 0.1?
A) After 0.1 hour, the mass of carbon-11 in the sample is 2g.
B) After 2 hours, the mass of carbon-11 in the sample is 0.1g.
C) After 0.1 hour, the mass of carbon-11 in the sample has decreased by exactly 2g.
D) After 2 hours, the mass of carbon-11 in the sample has decreased by exactly 0.1g.
Answer Explanation
In f(x), the input x represents hours and the output represents mass in grams. So f(2) = 0.1 means that after 2 hours, the mass is 0.1 grams — not a change of 0.1 grams, but the actual remaining mass.
The correct answer is B.
9Identifying an Increasing Exponential Table
Which table shows 3 values of x and their corresponding values of f(x) for an increasing exponential function f?
A)
x
f(x)
0
25
1
15
2
5
B)
x
f(x)
0
5
1
15
2
25
C)
x
f(x)
0
5
1
15
2
45
D)
x
f(x)
0
45
1
15
2
5
Answer Explanation
Look for a table where consecutive outputs share a constant ratio (not a constant difference) and the values are increasing.
Table C has values 5, 15, 45 — each term is exactly 3 times the one before it (a constant ratio of 3), and the values increase. The correct answer is C.
10Writing an Exponential Growth Model
According to a model, the population of Canada's Yukon Territory was 33 thousand in 2008 and increased by 2% of the previous year's population for each of the next 4 years. Which function represents this model, where f(x) is the population, in thousands, predicted by the model x years after 2008 and 0 ≤ x ≤ 4?
A) f(x) = 33(x)^1.02
B) f(x) = 33^(1.02x)
C) f(x) = 33(0.02)^x
D) f(x) = 33(1.02)^x
Answer Explanation
A 2% annual increase gives a growth factor of 1.02 per year, with 33 (thousand) as the initial value.
That gives f(x) = 33(1.02)^x. The correct answer is D.
11Interpreting a Function Value in Context
f(x) = 7.3(0.98)^x. The function shown models the annual soil erosion rate, in tons per acre year, on US cropland x years since 1982. According to the model, which is the best interpretation of 7.3(0.98)^10 in this context?
A) The number of acres of US cropland in 1982
B) The number of acres of US cropland in 1992
C) The annual soil erosion rate, in tons per acre per year, in 1982
D) The annual soil erosion rate, in tons per acre per year, in 1992
Answer Explanation
The expression 7.3(0.98)^10 is f(10), meaning x = 10 years since 1982 — that's the year 1992. The output of the function represents the soil erosion rate.
So this expression represents the annual soil erosion rate in 1992. The correct answer is D.
12Interpreting a Function Value in Context
The predicted number, B(t), of a certain type of bacterial cell in a growth medium is given by the function B(t) = 40(3)^t, where t is the time, in days, since the number of bacterial cells was first measured. Which of the following is the best interpretation of B(5) = 9,720 in this context?
A) 5 days after the number of bacterial cells was first measured, the predicted number of bacterial cells was 9,720
B) 243 days after the number of bacterial cells was first measured, the predicted number of bacterial cells was 9,720
C) 9,720 days after the number of bacterial cells was first measured, the predicted number of bacterial cells was 5
D) 9,720 days after the number of bacterial cells was first measured, the predicted number of bacterial cells was 243
Answer Explanation
In B(t), the input t is measured in days, and the output is the predicted number of bacterial cells. So B(5) = 9,720 means 5 days after the first measurement, the predicted count was 9,720 cells.
The correct answer is A.
13Interpreting a Y-Intercept in Context
f(t) = 7,000(0.67)^t. The given function f models the number of catalogs a company sent to their customers at the end of each year, where t represents the number of years since the end of 1995, and 0 ≤ t ≤ 5. If y = f(t) is graphed, which of the following is the best interpretation of the y-intercept of the graph in this context?
A) The minimum estimated number of catalogs the company sent to their customers during 5 years is 1,411
B) The minimum estimated number of catalogs the company sent to their customers during the 5 years was 7,000
C) The estimated number of catalogs the company sent to their customers at the end of 1995 was 1,411
D) The estimated number of catalogs the company sent to their customers at the end of 1995 was 7,000
Answer Explanation
The y-intercept occurs at t = 0: f(0) = 7,000(0.67)⁰ = 7,000(1) = 7,000, representing t = 0, which corresponds to the end of 1995.
Since the function is decreasing over the domain 0 ≤ t ≤ 5, the value at t = 0 is actually the largest value in that interval, not the minimum — ruling out the "minimum" options regardless of their numbers. The correct answer is D.
14Writing an Exponential Decay Model
A model estimates that the initial number of photons in a beam is 800 when the beam reaches the surface of a certain material. The model also estimates that when the beam passes through this material the number of photons in the beam decreases by 50% for each 13 millimeters of the material the beam passes through. Which equation represents this model, where I is the estimated number of photons in the beam when the beam reaches a point x millimeters from the surface?
A) I = 800(0.5)^(x/13)
B) I = 800(0.5)^x
C) I = 800(2)^(x/13)
D) I = 800(13)^(x/2)
Answer Explanation
Losing 50% every 13 millimeters means the decay factor is 0.5, applied once for every 13 mm traveled — captured by putting x/13 in the exponent.
That gives I = 800(0.5)^(x/13). The correct answer is A.
15Solving for a Constant from an Intercept
The function f is defined by f(x) = a^x + b, where a and b are constants. In the xy-plane, the graph of y = f(x) has an x-intercept at (2, 0) and a y-intercept at (0, −255). What is the value of b?
Answer Explanation
Use the y-intercept directly: f(0) = a⁰ + b = 1 + b, and this equals −255.
So 1 + b = −255, giving b = −256.
16Interpreting a Function Value in Context
A savings account is opened with an initial deposit of $6,000. The amount of money in the account t years after the initial deposit is given by the function f(t) = 6,000(1.01)^(2t). Which of the following is the best interpretation of the statement "f(5) is approximately equal to 6,627.73" in this context?
A) 5 years after the initial deposit, the amount of money, in dollars, in the account is 6,627.73
B) Every 5 years, the amount of money, in dollars, in the account increases by 6,627.73
C) 5 years after the initial deposit, the amount of money, in dollars, in the account has increased by 6,627.73
D) Every 5 years, the amount of money, in dollars, in the account decreases by 6,627.73
Answer Explanation
In f(t), the input t is years since the deposit, and the output is the total dollar amount in the account — not a change or increase, but the account's actual balance.
So f(5) = 6,627.73 means the account holds $6,627.73 five years after the initial deposit. The correct answer is A.
17Writing an Exponential Growth Model
A certain neighborhood had a population of 1,130 in 2006. Each year for the next 5 years the population of the neighborhood increased by approximately 4% of the population of the previous year. Which of the following equations represents the population, N, of the neighborhood t years after 2006, where t ≤ 5?
A) N = 0.04(1,130)^t
B) N = 1,130(0.04)^t
C) N = 1,130(1.04)^t
D) N = 1.04(1,130)^t
Answer Explanation
A 4% annual increase gives a growth factor of 1.04, with 1,130 as the initial population.
That gives N = 1,130(1.04)^t. The correct answer is C.
18Applying an Exponential Decay Model
In a certain state, the population of pheasants is estimated each year by counting the number of pheasants observed along certain roads in the state. On average, each year from 2005 to 2015 the number of pheasants counted per mile of road decreased by 3.5% of the number of pheasants per mile of road the previous year. Based on this average, if there were 6.97 pheasants per mile of road in this state in 2005, which of the following best approximates the number of pheasants per mile of road in 2015?
A) 0.965(6.97)^10
B) 1.035(6.97)^10
C) 6.97(0.035)^10
D) 6.97(0.965)^10
Answer Explanation
A 3.5% annual decrease gives a decay factor of 1 − 0.035 = 0.965. From 2005 to 2015 is 10 years, so the exponent is 10, applied to the starting value 6.97.
That gives 6.97(0.965)^10 — the initial value multiplied by the decay factor, not the decay factor raised to a power times a coefficient. The correct answer is D.
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