Mathfolis

Competing Function Model Validation

Unit 2 · Exponential and Logarithmic Functions

What AP Precalc asks here

When two function families both look plausible, the deciding question is: does a model's residual pattern look like random scatter, or does it have systematic curvature? Random residuals indicate a good fit; curved or trending residuals indicate the wrong family. A held-out data point is the cleanest validation: fit on the rest, predict the held-out point, and see which model is closer.

Residual interpretation

Random scatter near 0
Model fits the family\text{Model fits the family}
Curved (smile or frown)
Wrong family\text{Wrong family}
Funnel widening
Variance grows with x\text{Variance grows with } x
AP Tip: Three points always have a polynomial that fits them perfectly. To distinguish exponential from quadratic, you need a fourth point.
Type 1

Read the residual pattern

Scan the residuals for trends. Random scatter near zero = good fit; systematic shape = wrong family.

Example 1
A linear model gives residuals −3,−1,2,3,2,−1,−3-3, -1, 2, 3, 2, -1, -3 at successive equally spaced points. What does the pattern indicate? (A) The linear model fits — residuals are random (B) The linear model misses curvature; try a nonlinear family (C) The variance is growing with x; transform y (D) The residuals are all positive; a constant shift is needed

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Type 2

Compare two candidate models

Predict the held-out point with each candidate. The smaller absolute residual wins.

Example 2
Both L(x)=5+2xL(x) = 5 + 2x and E(x)=5⋅1.3xE(x) = 5 \cdot 1.3^x fit data up to x=3x = 3. At x=5x = 5, the observed value is 18.418.4. Which model fits better at x=5x = 5? (A) The linear model (B) The exponential model (C) They are equally close (D) Cannot tell from the given information

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Type 3

Reject a fit that misses one point

Compute both models through the first 3 points. Whichever matches the 4th data point is the right family.

Example 3
Given (0,1),(1,2),(2,4),(3,8)(0, 1), (1, 2), (2, 4), (3, 8), which single family fits all four points? (A) Linear (B) Quadratic (C) Exponential (D) Both quadratic and exponential

Practice more of this type— AI-generated · always-new problems

Generate Problems →