๐Ÿ  Homeโ€บ AP Precalculusโ€บ Unit 1โ€บ 1.13 Notes
๐Ÿ“Š Topic 1.13 ยท Unit 1

Modeling
Functions & Applications

Regression ยท linear, quadratic, cubic ยท TI-Nspire steps ยท residuals ยท choosing the right model

๐Ÿ“‹ Notes ๐Ÿƒ Flashcards ๐Ÿง  Quiz
๐Ÿ“Œ What Is Regression?
๐Ÿ“Š

The Big Idea

Modeling (also called regression) is a major AP Precalculus topic. We find a best-fit function that approximates the relationship in real-world data. In Unit 1 we work with polynomial models (linear, quadratic, cubic, quartic) and rational function models. These topics use the graphing calculator heavily and always involve context.

๐Ÿงฎ

Two Steps to Build Any Regression Model

Step 1: Enter the data into the graphing calculator
Step 2: Select the regression type and read off the equation

๐Ÿ“ Choosing the Right Model
๐Ÿ“
Linear ยท Quadratic ยท Cubic
Use the real-world context AND the shape of the data
๐Ÿ“ Linear
y = a + bx
Context: constant rate (price per item, constant speed)
Data: roughly straight, steady increase or decrease
๐Ÿ“ˆ Quadratic
y = axยฒ + bx + c
Context: area (A = ฯ€rยฒ)
Data: one turning point โ€” rises then falls (or falls then rises)
ใ€ฐ๏ธ Cubic
y = axยณ + bxยฒ + cx + d
Context: volume (V = sยณ)
Data: two turning points, S-shaped curve
โญ

AP Exam Tip โ€” Always Justify with BOTH

Context reason and data reason = full credit. Example: "Quadratic, because area involves rยฒ [context] AND the y-values increase to a maximum then decrease โ€” one turning point [data]."

๐Ÿ”‘

Quick Memory Aid

Length / rate / cost per unit โ†’ Linear  ยท  Area / circles (rยฒ) โ†’ Quadratic  ยท  Volume / spheres (rยณ) โ†’ Cubic

๐Ÿงฎ TI-Nspire Calculator Steps
๐Ÿงฎ TI-Nspire โ€” Building a Regression Model Step by Step
1
Open a new document โ†’ 4: Add Lists & Spreadsheet
2
Title your columns (press Enter after each name). Put the independent variable (x) in column 1.
3
Enter all data values in the columns, matching what is given.
4
Scatterplot: Add a new page โ†’ 5: Add Data & Statistics โ†’ click to add x and y variables โ†’ observe the shape.
5
Regression equation: Menu โ†’ 4: Analyze โ†’ 6: Regression โ†’ choose type (e.g. 1: Linear y=mx+b)
6
Record the equation on paper (round coefficients to 3 decimal places).
7
Evaluate at a value: Add a new page โ†’ 1: Calculator โ†’ press var โ†’ choose stat.regeqn( ) โ†’ enter input โ†’ record answer.
8
Find r: Select stat.r from the var menu โ†’ press Enter โ†’ record the correlation coefficient r.
โœ๏ธ Worked Examples 1โ€“3
๐Ÿ“Œ Example 1 โ€” Baby Weight: Linear Regression

The age (in weeks) and weight (in kg) of 5 randomly selected babies are listed. A linear regression y = a + bx is used, where y = predicted weight and x = age in weeks.

t (age in weeks)456812
W(t) (weight in kg)4.24.44.85.15.7
a
Write the linear model: Enter data โ†’ Linear Regression on calculator
b
Predict weight at t = 10 weeks:
y = 3.545 + 0.185(10) = 3.545 + 1.850 โ‰ˆ 5.395 kg
c
Find age when weight = 5.3 kg:
3.545 + 0.185t = 5.3 โ†’ 0.185t = 1.755 โ†’ t = 1.755 รท 0.185 โ‰ˆ 9.49 weeks
Linear model: W(t) = 3.545 + 0.185t
๐Ÿ“Œ Example 2 โ€” Choose the Model Type for Each Context
a
Water balloons (radius vs. volume): Volume of a sphere involves rยณ โ†’ CUBIC because the context is volume.
b
Totino's pizzas (count vs. total price): Each pizza has a roughly constant price โ†’ constant rate of change โ†’ LINEAR
c
Sprinkler (radius vs. area): Area = ฯ€rยฒ โ†’ involves rยฒ โ†’ QUADRATIC
๐Ÿ“Œ Example 3 โ€” Identify Model Type from Data & Write Regression Equation
x00.40.91.21.72.22.93.4
y510.615.417.118.016.510.22.8
a
Identify model type: y-values increase to a maximum between x = 1.2 and x = 1.7, then decrease. One turning point, roughly symmetric โ†’ QUADRATIC
b
Write the regression equation: Enter data on calculator โ†’ Quadratic Regression
y โ‰ˆ โˆ’4.883xยฒ + 15.958x + 4.996
๐Ÿ“‰ Residuals
๐Ÿ“‰
Actual โˆ’ Predicted
How far off is the model from the real data point?
Residual = Actual Value โˆ’ Predicted Value
๐Ÿ’ก

Interpreting the Sign

Positive (+): Actual > Predicted โ†’ model underestimated
Negative (โˆ’): Actual < Predicted โ†’ model overestimated
Zero: Perfect prediction

๐Ÿ“Œ Example 4 โ€” Residual for the 5-Week-Old Baby (model from Example 1)
1
Actual value (from table): W(5) = 4.4 kg
2
Predicted value (from model y = 3.545 + 0.185x): y = 3.545 + 0.185(5) = 3.545 + 0.925 = 4.47 kg
3
Residual: 4.4 โˆ’ 4.47 = โˆ’0.07 kg
โœ“
Interpretation: The residual is negative โ†’ the model overestimated. The model predicted 4.47 kg but the actual weight was only 4.4 kg โ€” it was 0.07 kg too high.
Residual = โˆ’0.07 kg ยท The model overestimated the weight of the 5-week-old baby by 0.07 kg.
Residual Actual vs Predicted Model Did...
Positive (+)Actual > PredictedUnderestimated โ†‘
Negative (โˆ’)Actual < PredictedOverestimated โ†“
Zero (0)Actual = PredictedPerfect โœ“