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2.6 · Linear Model
When should you choose a linear model for a data set?
Think about the rate of change
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2.6 · Quadratic Model
When should you choose a quadratic model for a data set?
Think about the shape of the scatter plot
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2.6 · Exponential Model
When should you choose an exponential model for a data set?
Think about how outputs relate to each other
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2.6 · Residual Definition
What is a residual? Write the formula.
Residual = ? − ?
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2.6 · Residual Sign
A model predicts 5.34 but the actual value is 5.5. What is the residual and what does it mean?
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2.6 · Good Residual Plot
What does a residual plot look like when the model IS appropriate for the data?
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2.6 · Bad Residual Plot
What does a residual plot look like when the model is NOT appropriate?
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2.6 · Apply: Identify Model
Outputs: 11, 8.2, 5, 2.3, −1 for equally-spaced inputs. Which model is best?
Look at the pattern of change
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✓ Quadratic Model
When rates of change are increasing/decreasing at a relatively constant rate
Data follows a "U"-shaped pattern (or upside-down U). 2nd differences are constant. Form: y = ax² + bx + c.
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✓ Linear Model
When the data reveals a relatively constant rate of change
1st differences between outputs are approximately equal. Graph looks like a straight line. Form: y = a + bx.
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✓ Residual Definition
Residual = Actual Output − Predicted Output
Positive residual = model underestimated (actual was higher than predicted). Negative = model overestimated.
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✓ Exponential Model
When output values are roughly proportional — each output ≈ previous × constant ratio
Look for repeated multiplication. Form: y = a·bˣ. Graph curves upward (growth) or downward (decay).
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✓ Good Residual Plot
Points scattered randomly above and below zero — NO clear pattern
Random scatter means the model captures the data's trend well. No structure in the residuals is what we want.
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✓ Residual Sign
Residual = 5.5 − 5.34 = +0.16; model underestimated
Positive (actual > predicted) → underestimate. If predicted > actual, residual is negative → overestimate.
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✓ Apply: Linear
Linear
11 → 8.2 → 5 → 2.3 → −1. Differences ≈ −2.9 each time — a roughly constant decrease → Linear model.
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✓ Bad Residual Plot
Points show a clear pattern (curve, wave, or systematic trend)
A pattern means the model did not capture all the structure in the data → the wrong model type was chosen.
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2.6 · Apply: Identify Model
Outputs: 9, 3.95, 2.1, 4.2, 8.8 for equally-spaced inputs. Which model fits best?
Sketch the pattern
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2.6 · Apply: Identify Model
Outputs: 10, 5.2, 2.4, 1.3, 0.7 for x = 1,3,5,7,9. Which model fits best?
Check ratios: 5.2/10 ≈ 0.52, 2.4/5.2 ≈ 0.46 …
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2.6 · Context: Model Choice
Mr. Passwater models paint (quarts) for circles of radius r (feet). Which model type?
Area of a circle = πr²
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2.6 · Over vs Under
Mr. Passwater uses his model to buy paint. Should the model overestimate or underestimate? Why?
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✓ Apply: Exponential Decay
Exponential
10, 5.2, 2.4, 1.3, 0.7 — each output is roughly half the previous. Consistent ratio ≈ 0.5 → exponential decay.
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✓ Apply: Quadratic
Quadratic
9 → 3.95 → 2.1 → 4.2 → 8.8. Values decrease then increase — a U-shaped pattern → Quadratic model.
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✓ Over vs Under
Overestimate — he does not want to run out of paint
Underestimating means buying too little and not finishing. Overestimating means extra paint but the project is complete.
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✓ Context: Model Choice
Quadratic — paint needed depends on area = πr²
Area grows as the square of the radius → quadratic relationship (y = ax²), not linear or exponential.