12.3.A.1Multiple Choice
Which function demonstrates exponential growth in general form?
Select A, B, C, or D to check your answer.
Directions: Work each problem on your own paper first. For multiple choice, select A, B, C, or D. You will see right away whether you are correct. If you miss, you will see the most likely reason, and you can try again. For short response, type your answer, then compare it with the model solution. Questions 1–10 go through 2.3.A.1–A.5 in order; questions 11–20 mix them. Round sensibly: money to the cent, counts to the nearest whole number, other decimals to the nearest hundredth.
Which function demonstrates exponential growth in general form?
Select A, B, C, or D to check your answer.
The table shows an exponential function f(x) = abx. (a) Find a and b. (b) Classify the function as growth or decay. (c) Find f(5).
| x | 0 | 1 | 2 | 3 |
| f(x) | 6 | 9 | 13.5 | 20.25 |
For f(x) = 2·3x, which expression shows the factors applied to find f(4)?
Select A, B, C, or D to check your answer.
A bacteria culture starts with 500 bacteria and triples every hour. (a) Write f(t) for the number of bacteria after t hours. (b) Write the count after 4 hours as a product of factors and evaluate it. (c) Find f(1.5) and state the domain of f.
The function g(x) = −4(0.5)x is an exponential function. Which statement describes its graph?
Select A, B, C, or D to check your answer.
Let f(x) = 80(0.5)x. (a) Find the average rate of change on [0, 1], [1, 2], and [2, 3]. (b) Is f increasing or decreasing? Is its graph concave up or concave down? (c) Find the maximum and minimum of f on the closed interval [0, 3].
The table gives outputs of a function f at equally spaced inputs. The outputs are not proportional, but an additive transformation of f is. Which function models f?
| x | 0 | 1 | 2 | 3 |
| f(x) | 11 | 14 | 20 | 32 |
Select A, B, C, or D to check your answer.
The table gives outputs of f at equally spaced inputs. (a) Show that the outputs are not proportional. (b) Find a function of the form f(x) = abx + k. (c) What value do the outputs get close to as x increases?
| x | 0 | 1 | 2 | 3 | 4 |
| f(x) | 30 | 18 | 12 | 9 | 7.5 |
Let f(x) = −6·2x. Which pair of limits is correct?
Select A, B, C, or D to check your answer.
Let g(x) = 50(0.8)x. Some values are shown. (a) Find limx→∞ g(x). (b) Find limx→−∞ g(x). (c) Can g(x) ever equal 0? Explain.
| x | 0 | 10 | 20 | 40 |
| g(x) | 50 | 5.37 | 0.576 | 0.0066 |
A car is worth $24,000 when new. Each year it loses 12% of its value from the year before. Which function gives its value V after t years?
Select A, B, C, or D to check your answer.
For each function, say whether it is exponential growth, exponential decay, or neither, and explain. (a) f(x) = 9(1.04)x (b) g(x) = 0.2(0.6)x (c) h(x) = −2(5)x (d) k(x) = 3(−2)x
An exponential function f(x) = abx satisfies f(0) = 7 and f(3) = 56. Which function is f?
Select A, B, C, or D to check your answer.
Which statement is true for every exponential function in general form?
Select A, B, C, or D to check your answer.
Without graphing, say whether each function is increasing or decreasing, and whether its graph is concave up or concave down. (a) 3(2)x (b) 3(0.5)x (c) −3(2)x (d) −3(0.5)x
A collectible is worth $1,200 and its value grows 8% per year. (a) Write V(t) for the value after t years. (b) Find V(5). (c) Find the value after 2.5 years, and explain why this input makes sense for an exponential function.
The outputs 7, 9, 13, 21 are given for x = 0, 1, 2, 3. They are not proportional. What constant k can be subtracted from every output so that the new outputs are proportional?
Select A, B, C, or D to check your answer.
A cup of coffee sits in a 20°C room. Its temperature T (in °C) is recorded each minute. (a) Show that T is not proportional over equal intervals. (b) Subtract the room temperature and show the result is proportional. (c) Write T(t) and predict T(6).
| t (min) | 0 | 1 | 2 | 3 |
| T (°C) | 90 | 76 | 64.8 | 55.84 |
Which statement about f(x) = 5(0.3)x is true?
Select A, B, C, or D to check your answer.
An exponential function f(x) = abx with b > 0 satisfies f(1) = 12 and f(3) = 48. (a) Find b and a, and write f(x). (b) Classify f as growth or decay. (c) Is f increasing or decreasing, and is its graph concave up or down? (d) Find limx→∞ f(x) and limx→−∞ f(x).