Same average rate, different functions
Goal: check representations and justify a conclusion using average rates.
Estimated time: 12 minutes. Input numbers and function values are dimensionless. A rate measures change in output per unit of input.
Given model
For every real input in 0 <= x <= 6, the functions are defined by these rules:
Quadratic: f(x) = x² + 2x + 5
Linear: g(x) = 8x + 5
The formulas and Graph 1 are correct. Exactly one numerical entry in Table 1 is wrong; all its other entries are correct. Repair only that table entry. Keep the formulas and graph unchanged. Use the repaired table for Parts B and C.
Equivalent graph description and data
Both graphs start at (0, 5) and end at (6, 53). The solid curved graph f passes through (2, 13) and (4, 29). The dashed straight graph g passes through (2, 21) and (4, 37). The f graph lies below the g graph between the shared endpoints. These are exact graphs of the supplied formulas.
On small screens or with enlarged text, scroll the table sideways to see every column.
| x | f(x) | g(x) |
|---|---|---|
| 0 | 5 | 5 |
| 2 | 13 | 21 |
| 4 | 29 | 37 |
| 6 | 53 | 53 |
Equivalent graph description and data
Both graphs start at (0, 5) and end at (6, 53). The solid curved graph f passes through (2, 13) and (4, 29). The dashed straight graph g passes through (2, 21) and (4, 37). The f graph lies below the g graph between the shared endpoints. These are exact graphs of the supplied formulas.
| x | f(x) | g(x) |
|---|---|---|
| 0 | 5 | 5 |
| 2 | 13 | 21 |
| 4 | 29 | 37 |
| 6 | 53 | 53 |
Table 1
On small screens or with enlarged text, scroll the table sideways to see every column.
| x | f(x) | g(x) |
|---|---|---|
| 0 | 5 | 5 |
| 2 | 15 | 21 |
| 4 | 29 | 37 |
| 6 | 53 | 53 |
A. Audit and repair
Identify the function and input of the wrong entry, then write its correct value. Show a calculation from the appropriate formula and cite a coordinate from Graph 1 that confirms your repair.
B. Compare equal-interval rates
For a function h, the average rate of change on [a, b] is:
(h(b) − h(a)) ÷ (b − a)
Using the repaired table, complete the rates below. Show one full calculation for each function, including the input difference.
On small screens or with enlarged text, scroll the table sideways to see every column.
| Input interval | Average rate for f | Average rate for g |
|---|---|---|
| [0, 2] | __________ | __________ |
| [2, 4] | __________ | __________ |
| [4, 6] | __________ | __________ |
Describe how each function's average rates change from one interval to the next. Give the numerical change for each pattern. Explain why comparing these changes is meaningful for the equal-width intervals used here.
C. Test a conclusion
A reviewer claims: “The functions have the same average rate on [0, 6], so they are the same function throughout the domain.”
Calculate each function's average rate on [0, 6]. Then decide whether the conclusion is justified. Use one interior input as a counterexample if needed, and explain why endpoint information does or does not settle the claim.
Now consider a separate question: if the formulas and declared families had not been supplied, would the four table rows alone prove a function is linear or quadratic for every input in [0, 6]? Explain briefly. The supplied formulas remain authoritative in the actual model.
Original formative practice. This is not an official AP question or scoring guide.
Complete response printing was not prepared. Copy all answers before closing, or permit the included offline print script.