AP Precalculus: Change in Linear and Exponential Functions (Drill 1)

Question 1. The table below shows selected values of a function f.

x	f(x)
0	3
1	6
2	12
3	24

Which of the following best describes f?

• A) Linear, because the output values increase by 3 for each unit increase in x
• B) Linear, because the output values increase by 6 for each unit increase in x
• C) Exponential, because the output values increase by a constant amount for each unit increase in x for the given function
• D) Exponential, because the output values are multiplied by a constant factor for each unit increase in x ✓

Explanation: Choice D is correct. Check successive ratios: 6/3 = 2, 12/6 = 2, 24/12 = 2. The output values are multiplied by 2 for each unit increase in x, that's exponential behavior. Choice A is incorrect because the differences are not constant: 6 - 3 = 3, but 12 - 6 = 6 and 24 - 12 = 12. The value 3 is the initial output, not a common difference. Choice B is incorrect because although 6 is the second output value, the differences between consecutive outputs (3, 6, 12) are not constant, so f is not linear. Choice C is incorrect because the reasoning contradicts the correct function type: exponential functions are characterized by proportional (multiplicative) change, not constant additive change. Constant additive change is the hallmark of linear functions.

Question 2. The table below shows selected values of a function g.

x	g(x)
2	11
4	17
6	23
8	29

Which of the following could be an expression for g(x)?

• A) g(x) = 3x + 5 ✓
• B) g(x) = 5(3)^x
• C) g(x) = 6x − 1
• D) g(x) = 11(1.5)^x

Explanation: Choice A is correct. The differences between consecutive output values are constant: 17 - 11 = 6, 23 - 17 = 6, 29 - 23 = 6. However, the input values increase by 2 each time, so the rate of change is 6/2 = 3. This means g is linear with slope 3. Using the point (2, 11): g(x) = 11 + 3(x - 2) = 3x + 5. Verify: g(2) = 6 + 5 = 11 ✓, g(4) = 12 + 5 = 17 ✓. Choice B is incorrect because g(x) = 5(3)^x is exponential; g(2) = 5(9) = 45 ≠ 11, so it does not fit the table. Choice C is incorrect because g(x) = 6x - 1 gives g(2) = 11 ✓ but g(4) = 23 ≠ 17. This choice uses 6 as the slope rather than the correct rate of change of 3 (confusing the common difference of 6 with the slope, which must account for the input interval of 2). Choice D is incorrect because exponential functions are ruled out: the output values increase by a constant amount (6 per 2 units of input), not by a constant factor, so g is linear, not exponential.

Question 3. An exponential function f satisfies f(1) = 10 and f(3) = 40. Which of the following gives an expression for f(x)?

• A) f(x) = 5(2)^x-1
• B) f(x) = 10(2)^x-1 ✓
• C) f(x) = 10 + 15(x - 1)
• D) f(x) = 4(2.5)^x

Explanation: Choice B is correct. Since f is exponential, write f(x) = a·b^x. From f(1) = 10 and f(3) = 40: f(3)/f(1) = 40/10 = 4 = b^3-1 = b², so b = 2. Then f(1) = a·2¹ = 10, giving a = 5, so f(x) = 5(2)^x. Expressing this using the known point (1, 10) directly: f(x) = 10·2^x-1. Verify: f(1) = 10·2⁰ = 10 ✓, f(3) = 10·2² = 40 ✓. Choice A is incorrect because f(x) = 5(2)^x-1 gives f(1) = 5(2)⁰ = 5 ≠ 10 and f(3) = 5(2)² = 20 ≠ 40; subtracting 1 from the exponent divides the entire function by 2, shifting it off both given values. Choice C is incorrect because f(x) = 10 + 15(x - 1) is linear with constant rate of change 15, not exponential; it passes through (1, 10) and (3, 40) but cannot be the answer since the problem specifies f is exponential. Choice D is incorrect because f(x) = 4(2.5)^x gives f(1) = 10 ✓ but f(3) = 4(15.625) = 62.5 ≠ 40; this base was not determined from the two given points.

Question 4. A savings account earns interest so that its balance doubles every 12 years. A checking account grows by $500 per year. Both accounts currently have a balance of $2,000. Which of the following statements correctly compares the two accounts over a long period of time?

• A) The checking account will always have a higher balance because it grows by a fixed amount each year.
• B) The savings account will eventually have a higher balance because exponential growth will outpace linear growth over time. ✓
• C) The two accounts will always have the same balance because they start at the same value.
• D) The savings account will always have a higher balance because it grows multiplicatively.

Explanation: Choice B is correct. The savings account grows exponentially (doubling every 12 years), while the checking account grows linearly ($500 per year). In the short term, the linear account may pull ahead, after 12 years, the checking account has 2000 + 500(12) = $8,000, while the savings account has only $4,000. But exponential growth always eventually overtakes linear growth. The exponential function accelerates while the linear function adds the same amount every year. Choice A is incorrect because while the checking account may be larger in the short run, linear growth is eventually overtaken by exponential growth, the question specifies "over a long period of time." Choice C is incorrect because starting at the same value does not mean the accounts remain equal; the two functions have different growth patterns and diverge immediately. Choice D is incorrect because "always" is too strong. In the early years the checking account actually grows faster in absolute dollar terms. The savings account only surpasses the checking account after a crossover point.

Question 5. The table below shows values of two functions, f and g, at selected inputs.

x	f(x)	g(x)
0	4	4
1	7	8
2	10	16
3	13	32

Which of the following statements about f and g is true?

• A) Both f and g are linear because both are increasing.
• B) f is exponential with base 3, and g is linear with slope 4.
• C) f is linear with slope 3, and g is exponential with base 2. ✓
• D) Both f and g are exponential because both start at the same initial value.

Explanation: Choice C is correct. For f: differences are 7 - 4 = 3, 10 - 7 = 3, 13 - 10 = 3. The output values increase by a constant 3 for each unit increase in x, so f is linear with slope 3. For g: ratios are 8/4 = 2, 16/8 = 2, 32/16 = 2. The output values are multiplied by a constant factor of 2 for each unit increase in x, so g is exponential with base 2. The function g can be written as g(x) = 4(2)^x. Choice A is incorrect because being increasing is not sufficient to conclude linearity, exponential functions are also increasing. Function type is determined by the pattern of change (constant differences vs. constant ratios), not direction of change. Choice B is incorrect because the descriptions of f and g are swapped and the values are wrong: f has constant differences (linear), not constant ratios; checking f: 7/4 ≠ 10/7, confirming f is not exponential. Choice D is incorrect because having the same initial value (f(0) = g(0) = 4) does not determine function type. The pattern of change, not the starting value, distinguishes linear from exponential functions.