π Big Picture
On the AP Exam you must determine which model β linear, quadratic, or exponential β best fits a data set. Use the patterns in the data table and the shape of the scatter plot to decide.
π Comparing the Three Model Types
βοΈ Example 1 β Identify Each Model
Linear
| x | f(x) |
|---|---|
| 0 | 11 |
| 2 | 8.2 |
| 4 | 5 |
| 6 | 2.3 |
| 8 | β1 |
Decreasing at ~constant rate β Linear
Quadratic
| x | g(x) |
|---|---|
| β1 | 2 |
| 1.5 | 5.5 |
| 4 | 10.5 |
| 6.5 | 5.75 |
| 9 | 2.25 |
Goes up then back down β U-shape β Quadratic
Exponential
| x | h(x) |
|---|---|
| 1 | 10 |
| 3 | 5.2 |
| 5 | 2.4 |
| 7 | 1.3 |
| 9 | 0.7 |
Outputs roughly halving each step β Exponential
Exponential
| x | k(x) |
|---|---|
| β2 | 2 |
| 1 | 3 |
| 4 | 4.5 |
| 7 | 6.75 |
| 10 | 10.25 |
Outputs multiply by ~1.5 each 3 steps β Exponential
π Residuals
A residual is the difference between the actual and predicted output:
Residual = Actual β Predicted
Positive β model underestimated | Negative β model overestimated
βοΈ Example 2 β Baby Weight Linear Model
W(t) = 3.34 + 0.8t
| t | W(t) |
|---|---|
| 0 | 3.2 |
| 1 | 4.2 |
| 2 | 5.1 |
| 3 | 5.8 |
| 4 | 6.4 |
(a) LinReg β W(t) = 3.34 + 0.8t
(b) Predict at t = 2.5:
W(2.5) = 3.34 + 0.8(2.5) = 5.34 kg
(c) Actual = 5.5 kg
Residual = 5.5 β 5.34 = 0.16
Positive β model underestimated
(b) Predict at t = 2.5:
W(2.5) = 3.34 + 0.8(2.5) = 5.34 kg
(c) Actual = 5.5 kg
Residual = 5.5 β 5.34 = 0.16
Positive β model underestimated
π Residual Plots β Is the Model Appropriate?
Key Rule: A good residual plot shows NO pattern β points scattered randomly above and below zero. A clear pattern means the wrong model was chosen.
Linear Residuals
β οΈ Pattern β bad fit
Quadratic Residuals
β οΈ Pattern β bad fit
Exponential Residuals
β No pattern β good fit
βοΈ Example 3 β Interpreting One Residual Plot
Residual plot shows random scatter (no pattern). Best conclusion?
(A) Not appropriate β no pattern β
(B) Not appropriate β shows pattern β
(C) Appropriate β residuals show no pattern β
(D) Appropriate β shows a pattern β
(B) Not appropriate β shows pattern β
(C) Appropriate β residuals show no pattern β
(D) Appropriate β shows a pattern β
Answer: C. No pattern = the model fits well.
βοΈ Example 4 β Three Residual Plots Side by Side
Linear and quadratic plots showed clear patterns. Exponential plot showed no pattern. Best model?
Exponential model β because its residual plot shows no pattern (random scatter above and below zero), indicating it captures the data's trend most appropriately.
βοΈ Example 5 β Context-Based Model Choice
Mr. Passwater's Paint Problem
Modeling paint (quarts) needed for circles of radius r (feet).
(a) Best model: Quadratic
Paint needed β Area = ΟrΒ² β a quadratic relationship.
(b) Should the model overestimate
Reason: He does not want to run out of paint mid-project. Overestimating means he buys enough to finish.
Paint needed β Area = ΟrΒ² β a quadratic relationship.
(b) Should the model overestimate
Reason: He does not want to run out of paint mid-project. Overestimating means he buys enough to finish.
Exam tip: For over vs underestimate questions, think about the real-world consequence of having too little.
Card 1 of 6
Concept
When should you use a linear model?
Tap to reveal β¨
Answer
When the data shows a relatively constant rate of change
1st differences are approximately equal. Graph looks like a straight line. Form: y = a + bx.
Topic 2.6
Competing Function Model Validation
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Question 1 of 5
Outputs 1, 2.6, 5, 6.4, 8.7 for equally-spaced inputs. Which model is most appropriate?
Question 2 of 5
Outputs 9, 3.95, 2.1, 4.2, 8.8 for equally-spaced inputs. Which model is most appropriate?
Question 3 of 5
W(2.5) = 5.34 kg predicted; actual weight = 5.5 kg. What is the residual and what does it mean?
Question 4 of 5
An exponential regression's residual plot shows points scattered randomly above and below zero with no pattern. Best conclusion?
Question 5 of 5
Mr. Passwater models paint (quarts) needed for circular murals of radius r. Which model type and estimation direction?
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Keep going!