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.
| x | f(x) | g(x) |
|---|---|---|
| 0 | 5 | 5 |
| 2 | 13 | 21 |
| 4 | 29 | 37 |
| 6 | 53 | 53 |
Table 1
| 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.
| 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.