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1.13 · What is regression?
What is a regression model? What is it used for?
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1.13 · Two steps to build a model
What are the two steps to build a regression model on a graphing calculator?
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1.13 · Linear model — formula
What is the general form of a linear regression model?
y = a + bx
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1.13 · Quadratic — context clue
Give a real-world context that calls for a QUADRATIC model. Why?
Think: area
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1.13 · Cubic — context clue
Give a real-world context that calls for a CUBIC model. Why?
Think: volume
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1.13 · Sprinkler problem
A sprinkler waters a circular area. Radius vs. area watered. Which model?
Area = πr²
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1.13 · Data shape → quadratic
Data shows y-values increase to a maximum near x = 1.5 then decrease. Which model?
Count turning points
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1.13 · Residual formula
State the formula for a residual.
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✓ Two steps to build a model
Step 1: Enter the data into the calculator. Step 2: Select the regression model type and read off the equation.
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✓ What is regression?
A regression model is a best-fit function that approximates the relationship between two real-world quantities. It's not exact, but close enough to be useful for predictions.
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✓ Quadratic — context clue
Area context — e.g. A = πr². As radius doubles, area quadruples (r²). Any context involving AREA calls for a quadratic model.
Example: sprinkler, circle area
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✓ Linear model — formula
y = a + bx (or y = mx + b). A constant rate of change — every 1 unit increase in x changes y by a fixed amount b.
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✓ Sprinkler problem
QUADRATIC. Area = πr² → relationship involves r² → quadratic.
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✓ Cubic — context clue
Volume context — e.g. V = s³ or V = (4/3)πr³. Any context involving VOLUME calls for a cubic model.
Example: water balloon, sphere
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✓ Residual formula
Residual = Actual Value − Predicted Value. Measures how far off the model is from a real data point.
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✓ Data shape → quadratic
QUADRATIC. One turning point (peak or valley) is the signature of a quadratic. Data is also roughly symmetric about the turning point.
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1.13 · Positive residual
A residual is +0.3 kg. What does this mean about the model?
Actual − Predicted = +0.3
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1.13 · Negative residual — Example 4
Residual = −0.07 kg (baby at 5 weeks). Interpret fully.
Actual = 4.4, Predicted = 4.47
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1.13 · Forward prediction
Model: W(t) = 3.545 + 0.185t. Predict weight at t = 10 weeks.
Substitute t = 10
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1.13 · Inverse prediction
Model: W(t) = 3.545 + 0.185t. A baby weighs 5.3 kg. Predict the age.
Set = 5.3, solve for t
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1.13 · Quadratic regression equation
Data gives y = −4.883x² + 15.958x + 4.996. What does the negative leading coefficient tell you?
Parabola opens...
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1.13 · Volume vs Area vs Rate
Which model goes with volume? Area? Constant rate?
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1.13 · AP tip — justify model
On the AP exam, how must you justify your choice of model type?
Two things needed
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1.13 · TI-Nspire — find regression
After entering data, what TI-Nspire menu path gives the regression equation?
Menu → Analyze → ...
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✓ Negative residual — Example 4
Residual = 4.4 − 4.47 = −0.07 kg. Negative → model OVERESTIMATED. The prediction was 0.07 kg too high.
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✓ Positive residual
Residual = +0.3 → Actual > Predicted → model UNDERESTIMATED. The real value was 0.3 kg higher than the prediction.
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✓ Inverse prediction
3.545 + 0.185t = 5.3 → 0.185t = 1.755 → t ≈ 9.49 weeks.
Set model = given output, solve for input.
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✓ Forward prediction
W(10) = 3.545 + 0.185(10) = 3.545 + 1.850 = 5.395 kg
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✓ Volume vs Area vs Rate
Volume → CUBIC (r³) · Area → QUADRATIC (r²) · Constant rate → LINEAR
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✓ Quadratic regression equation
Negative leading coefficient (−4.883) → parabola opens DOWNWARD → a maximum exists. The model rises to a peak then falls.
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✓ TI-Nspire — find regression
Menu → 4: Analyze → 6: Regression → choose type (1: Linear, 2: Quadratic, etc.)
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✓ AP tip — justify model
Must include: (1) CONTEXT reason (e.g. 'area involves r²') AND (2) DATA reason (e.g. 'one turning point'). Both = full credit.