12.5.A.1Multiple Choice
The outputs of four functions at x = 0, 1, 2, 3 are listed. Which function could be exponential?
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 1–19 go through 2.5.A.1 to 2.5.B.3 in order, and question 20 uses several. A graphing calculator is allowed for regression and for solving. Round sensibly: money to the cent, counts of people or cells to the nearest whole number, other decimals to the nearest hundredth.
The outputs of four functions at x = 0, 1, 2, 3 are listed. Which function could be exponential?
Select A, B, C, or D to check your answer.
The table gives f at equally spaced inputs. (a) Show that the outputs are proportional. (b) Write f(x) = abx. (c) Is it growth or decay, and what is the percent change per step? (d) Find f(5).
| x | 0 | 1 | 2 | 3 |
| f(x) | 80 | 60 | 45 | 33.75 |
The outputs 12, 14, 18, 26 are given at x = 0, 1, 2, 3. They are not proportional. What constant can be subtracted from every output to make the new outputs proportional?
Select A, B, C, or D to check your answer.
A drink's temperature T (in °C) is recorded each minute in a cool room. (a) Show that T is not proportional. (b) Subtract 10 from each output and show the result is proportional. (c) Write T(t). (d) Find T(6) and say what temperature T approaches.
| t (min) | 0 | 1 | 2 | 3 | 4 |
| T (°C) | 30 | 20 | 15 | 12.5 | 11.25 |
An exponential function f(x) = abx has f(2) = 12 and f(4) = 48. Which is f(x)?
Select A, B, C, or D to check your answer.
An exponential function f has f(1) = 10 and f(4) = 1250. (a) Find b. (b) Find a and write f(x). (c) Find f(0) and f(3).
Which statement describes the function f(x) = −2·3x + 8?
Select A, B, C, or D to check your answer.
A drink starts at 30°C in a 6°C room. Each hour the gap between its temperature and 6°C shrinks by 20%. (a) What is the base? (b) Construct T(t) by transforming the parent function. (c) Find T(3). (d) What does T approach as t increases?
An exponential regression on a data set gives y = 150(1.08)x, where x is in years. Which interpretation is correct?
Select A, B, C, or D to check your answer.
A town's population (in thousands) is recorded each year. Exponential regression gives y ≈ 11.948(1.0948)x. (a) State the initial value and the percent growth per year. (b) Predict the population in year 8. (c) Is that an interpolation or an extrapolation? How confident should you be?
| year | 0 | 1 | 2 | 3 | 4 | 5 |
| thousands | 12.0 | 13.1 | 14.2 | 15.7 | 17.1 | 18.9 |
A population is modeled by P(t) = 300e0.05t. Which is an equivalent form a·bt?
Select A, B, C, or D to check your answer.
A medicine is modeled by A(t) = 50e−0.1t mg after t hours. (a) Rewrite A as a·bt. (b) What percent is removed each hour? (c) Find A(10).
A quantity decreases by 7% each year. Which is the growth factor b in the model f(x) = abx?
Select A, B, C, or D to check your answer.
A value is modeled by V(t) = 900(1.2)t, where t is in years. Which statement is true?
Select A, B, C, or D to check your answer.
A town has 8,000 people and grows 4% per year. (a) Write P(t). (b) Find P(5). (c) Find the increase in people during year 1 and during year 2. Why are they different if the growth is always 4%?
A quantity is modeled by Q(t) = 3t, where t is in hours. (a) Rewrite Q to show the factor every 2 hours. (b) Rewrite Q to show the factor every 6 hours. (c) Check both forms at t = 12.
A quantity is modeled by f(d) = 2d, where d is days. Which equivalent form shows the factor every 3 days?
Select A, B, C, or D to check your answer.
A colony is modeled by P(t) = 200(1.5)t, where t is the number of days since the colony was first measured. Which input does not make sense in this context?
Select A, B, C, or D to check your answer.
An account is modeled by A(t) = 2000(1.05)t dollars after t years. (a) Find A(8). (b) Use a graph or table to find when the balance reaches $3,000. (c) If interest is credited once a year, after how many whole years does the balance first exceed $3,000?
A culture has 18 thousand cells at hour 2 and 486 thousand cells at hour 5. Assume exponential growth. (a) Find a model N(t) = abt. (b) Find the percent growth per hour. (c) Find N(3.5). (d) Find the growth factor every 2 hours. (e) Use a graph to estimate when the culture reaches 1,000 thousand.