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3.M Focus on Modeling (Chapter Supplement)

Fit polynomial models to data and interpret predictions with realistic constraints.

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Ideas in this lesson

Explore the lesson brief and the worked example for Idea $1$ . Create a free account to continue with the remaining ideas and practice.

01Choose polynomial degree and compare least-squares fitWorked steps · five questions 02Fit quadratic data and interpret an optimumWorked steps · five questions 03Fit cubic growth data and invert the modelWorked steps · five questions 04Model projectile data with two inverse predictionsWorked steps · five questions 05Model drainage with a physically limited quadraticWorked steps · five questions 06Evaluate residuals, interpolation, and extrapolationWorked steps · five questions

Lesson brief and rules

3.M Focus on Modeling (Chapter Supplement)

Inspect a scatterplot, choose a justified polynomial degree, fit the model, and test predictions against the data and physical domain.

Remember
A model must respect the observed data and the practical meaning of its domain.

Worked example

1

01 / In a polynomial trend study, the following invented observations are recorded as input-output pairs:

(0,3.23)

(1,4.54)

(2,6.03)

(3,7.4)

(4,8.77)

(5,10.26)

(6,11.57)

. Keep every observation, including repeated inputs. Use technology to obtain the requested least-squares fit. Show a labeled scatter plot and overlay your fitted curve. Compare linear, quadratic, and cubic fits by their sums of squared residuals. Choose the lowest degree whose

SSE

is below

0.1

, describe whether the data require a peak or valley, and explain why merely choosing the smallest

SSE

can overfit. If each measured output is increased by

2

, explain exactly how the fitted coefficients and

SSE

change without refitting. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

02 / Enter the

7

observations as paired lists; do not sort the two lists independently.

03 / Use the calculator polynomial regression command of degree

1 (

not interpolation

)

, then store its coefficients. This minimizes

\sum_i(y_i-p(x_i))^{2}

. The normal equations are

(A^{T}A)c=A^{T}y

, where row

i

A

(1,x_i)

04 / The fitted model is

p(x)=1.4x+3.2

. Coefficients displayed to six decimals; retain full precision in calculations.

05 / Overlay this curve on the data for

0\le x\le 6

. The vertical residuals, in input order, are

0.0300, -0.0600, 0.0300, 0.0000, -0.0300, 0.0600, -0.0300

; their squared sum is

0.010800

06 / degree

1

SSE=0.010800

p(x)=1.4x+3.2

; degree

2

SSE=0.010800

p(x)=1.4x+3.2

; degree

3

SSE=0.008400

p(x)=-0.003333x^{3}+0.03x^{2}+1.333333x+3.22

07 / The stipulated

SSE

rule selects degree

1

. Higher-degree models contain the lower-degree family and therefore cannot have larger best-fit

SSE

; this alone does not establish a better physical model.

08 / Adding

2

to every output adds

2

only to the fitted constant term. Every residual and the

SSE

stay unchanged, since observed and predicted outputs shift equally.

09 /

Reference scatter plot and fitted polynomial. Open circles: observations; solid curve: fitted model.

Worked example 2 · Choose polynomial degree and compare least-squares fit

In a polynomial trend study, the following invented observations are recorded as input-output pairs: $(0,2.435)$ , $(1,5.73)$ , $(2,8.035)$ , $(3,9.0)$ , $(4,8.765)$ , $(5,7.47)$ , $(6,4.765)$ . Keep every observation, including repeated inputs. Use technology to obtain the requested least-squares fit. Show a labeled scatter plot and overlay your fitted curve. Compare linear, quadratic, and cubic fits by their sums of squared residuals. Choose the lowest degree whose $SSE$ is below $0.1$ , describe whether the data require a peak or valley, and explain why merely choosing the smallest $SSE$ can overfit. If all outputs are converted to a unit half as large, explain how coefficients and $SSE$ change. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

Enter the $7$ observations as paired lists; do not sort the two lists independently.
Use the calculator polynomial regression command of degree $2 ($ not interpolation $)$ , then store its coefficients. This minimizes $\sum_i(y_i-p(x_i))^{2}$ . The normal equations are $(A^{T}A)c=A^{T}y$ , where row $i$ of $A$ is $(1,x_i,x_i^{2})$ .
The fitted model is $p(x)=-0.6x^{2}+4x+2.4$ . Coefficients displayed to six decimals; retain full precision in calculations.
Overlay this curve on the data for $0\le x\le 6$ . The vertical residuals, in input order, are $0.0350, -0.0700, 0.0350, 0.0000, -0.0350, 0.0700, -0.0350$ ; their squared sum is $0.014700$ .
degree $1$ : $SSE=30.254700$ , $p(x)=0.4x+5.4$ ; degree $2$ : $SSE=0.014700$ , $p(x)=-0.6x^{2}+4x+2.4$ ; degree $3$ : $SSE=0.011433$ , $p(x)=-0.003889x^{3}-0.565x^{2}+3.922222x+2.423333$
The stipulated $SSE$ rule selects degree $2$ . Higher-degree models contain the lower-degree family and therefore cannot have larger best-fit $SSE$ ; this alone does not establish a better physical model.
The numerical outputs double. Every fitted coefficient doubles and every residual doubles, so the $SSE$ is multiplied by $4$ .

Reference scatter plot and fitted polynomial. Open circles: observations; solid curve: fitted model.

Written-work review:

Enter every data pair correctly and state the specified regression degree.
Produce a labeled scatter plot with the fitted curve overlaid; the coordinates and fitted equation in the solution specify the reference graph.
Show the requested numerical work with units, retain full regression precision until final rounding, and justify the physical interval or prediction limitations.

15 questions with a worked answer for each.Start practice ↗

Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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3.M Focus on Modeling (Chapter Supplement)

Ideas in this lesson

3.M Focus on Modeling (Chapter Supplement)