12.6.A.1Multiple Choice
The outputs of four functions at x = 0, 1, 2, 3, 4 are listed. Which function could be modeled by a quadratic function?
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 1–18 go through 2.6.A.1 to 2.6.B.2 in order, and questions 19–20 mix them. 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, 4 are listed. Which function could be modeled by a quadratic function?
Select A, B, C, or D to check your answer.
The table gives f at equally spaced inputs. (a) Decide whether f is linear, quadratic, or exponential, and show the evidence. (b) Write a model. (c) Find f(6).
| x | 0 | 1 | 2 | 3 | 4 |
| f(x) | 81 | 54 | 36 | 24 | 16 |
The outputs 50, 44, 38, 32, 26 are given at x = 0, 1, 2, 3, 4. Which type of model do they suggest?
Select A, B, C, or D to check your answer.
The table gives a data set. (a) Find the ratios of consecutive outputs. Are they close to constant? (b) Use exponential regression to write a model. (c) What is the percent change per step? (d) Predict y at x = 6.
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| y | 50.2 | 36.8 | 28.1 | 20.6 | 15.7 | 11.6 |
A data set is shown. (a) Find the first and second differences. What type of model do they suggest? (b) Use regression to write a model. (c) Predict y at x = 7.
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| y | 1.2 | 3.6 | 8.9 | 16.1 | 26.0 | 38.2 |
The height of a diver above the water is recorded every 0.2 seconds after she leaves the platform. The heights rise, reach a maximum, and then fall. Which type of model is most appropriate?
Select A, B, C, or D to check your answer.
Which situation is most appropriately modeled by an exponential function?
Select A, B, C, or D to check your answer.
A gym records its number of members at the end of each month. (a) Do the first differences or the ratios look more constant? (b) Choose a model type and justify it with the context that members join at a steady pace. (c) Use regression to write the model and predict the number of members at month 12.
| month | 0 | 1 | 2 | 3 | 4 | 5 |
| members | 40 | 52 | 63 | 77 | 88 | 100 |
A rock is launched upward from a cliff. The table shows its height h (in feet) at time t (in seconds). (a) Which model type is appropriate and why? (b) The quadratic model is h(t) = −16t² + 48t + 6. Find the maximum height and when it occurs. (c) The model gives h(5) = −154. Explain why the model should not be used for t = 5.
| t (s) | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
| h (ft) | 6 | 26 | 38 | 42 | 38 | 26 | 6 |
Which expression gives the residual of a model at a data point?
Select A, B, C, or D to check your answer.
The model y = 2x + 5 is used for the data below. (a) Find the predicted values and the residuals. (b) Describe the residual plot. (c) Is the model justified? Explain.
| x | 0 | 1 | 2 | 3 | 4 |
| actual | 5.2 | 6.8 | 9.1 | 10.8 | 13.1 |
A line is fit to a data set and the residual plot forms a clear U shape. What is the best conclusion?
Select A, B, C, or D to check your answer.
Four residual plots are shown for four different models of different data sets. Which plot supports the claim that its model is justified?
Select A, B, C, or D to check your answer.
The data from Question 5 is modeled by a line and by a quadratic. The residuals are shown. (a) Describe each residual plot. (b) Which model is justified? (c) Which model would you use to predict y at x = 7, and why?
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| line resid. | +4.06 | −0.95 | −3.06 | −3.27 | −0.78 | +4.00 |
| quad resid. | +0.05 | −0.15 | +0.15 | −0.06 | +0.02 | −0.01 |
At x = 4 a model predicts 30 but the actual value is 27. Which statement is correct?
Select A, B, C, or D to check your answer.
The model y = 1.5x + 2 is used for the data below. (a) Find the signed errors (residuals). (b) Find the unsigned errors and their mean. (c) At which inputs does the model underestimate?
| x | 0 | 1 | 2 | 3 | 4 |
| actual | 2.4 | 3.4 | 5.3 | 6.2 | 8.6 |
A caterer uses a model to estimate the number of meals to prepare. Running short is much worse than having extra. Which residuals would the caterer prefer?
Select A, B, C, or D to check your answer.
A venue uses the model y = 4x + 6 to estimate crowd size (in hundreds) on day x, and safety rules say it must never underestimate. The data is shown. (a) Find the residuals. (b) On which inputs does the model underestimate? (c) By how much could you raise the model so that it never underestimates? (d) What does the residual pattern suggest about the type of model?
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| actual | 8.4 | 9.6 | 12.5 | 16.1 | 21.4 | 27.0 | 34.3 |
An exponential model for a viral video has a residual plot with no pattern on days 0 to 6. Which statement is the most reasonable?
Select A, B, C, or D to check your answer.
The number of times a hashtag is used each day is recorded. Each person who sees it passes it along to others. (a) Which type of model does the context suggest? Check with the differences and ratios. (b) Use regression to write the model and predict day 9. (c) The residuals for the exponential model and for a linear model are shown. Which model is justified? (d) For the exponential model, on which days does it underestimate?
| day | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| uses | 3.1 | 4.1 | 6.0 | 8.1 | 11.6 | 16.0 | 22.7 |
| exp. resid. | +0.07 | −0.13 | +0.09 | −0.15 | +0.08 | −0.10 | +0.22 |
| linear resid. | +2.32 | +0.17 | −1.08 | −2.13 | −1.78 | −0.53 | +3.02 |