Desmos: Regression and Checking Models

17.2: Desmos: Regression and Checking Models. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 2 of 63 parts Practice Flashcards

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Linear regression setup

Objective: Relate a table and linear regression parameters to an exact small example.

Prerequisite skills: scatterplots, model families, and slope. Put observations into x₁,y₁ table columns. A regression relation such as y₁ ~ mx₁+b asks Desmos to fit m and b; use the tilde for fitting rather than an equality defining a new graph. If the data lie exactly on a line, algebra should reproduce its parameters. Common error: using x or y instead of the named table columns in the fit relation. Restrictions and interpretation: Regression describes a fit, not an exact law or proof of cause; inspect units, residuals, and the observed input range.

Formulas

linear regression relation: y_1 ~ m x_1 + b

Worked examples

Try itA data table contains (1,30), (2,53), and (4,99). It lies exactly on y = mx + b. What slope should linear regression return?Show answer

Using the first two points gives m=(53−30)/(2−1)=23. The third point follows the same line. Regression should recover this exact slope up to display precision. Therefore the requested answer is 23.

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Linear regression setup

Objective: Relate a table and linear regression parameters to an exact small example.

Formulas

linear regression relation: y_1 ~ m x_1 + b

Worked examples

Try itA data table contains (1,30), (2,53), and (4,99). It lies exactly on y = mx + b. What slope should linear regression return?Show answer

Choose a nonlinear model family

Objective: Use differences or ratios to choose a regression form.

A quadratic regression can use y₁ ~ ax₁²+bx₁+c, while an exponential regression can use y₁ ~ ab^(x₁) for suitable positive outputs. Choose a family based on structure and residuals rather than a high fit statistic alone. Exact constructed data permit an algebraic check. Common error: choosing a model solely because its graph looks close over a tiny window. Restrictions and interpretation: Regression describes a fit, not an exact law or proof of cause; inspect units, residuals, and the observed input range.

Formulas

quadratic: y_1 ~ ax_1²+bx_1+c; exponential: y_1 ~ ab^(x_1)

Worked examples

Try itOutputs at x = 0,1,2,3 are 25, 75, 225, 675. For y = ab^x, what should b be?Show answer

The consecutive ratios are all 3. The value at x=0 gives coefficient a=25; the common growth factor is b=3. An exponential regression using these exact data should reproduce it. Therefore the requested answer is 3.

Residual and model checking

Objective: Compare a fitted prediction with observed data.

After fitting, compute observed-minus-predicted residuals in the original output units. A systematic curve in residuals can signal a poor model family. An out-of-range prediction can be unreliable even when in-range residuals are small. Common error: reversing the residual sign or treating fit as a causal mechanism. Restrictions and interpretation: Regression describes a fit, not an exact law or proof of cause; inspect units, residuals, and the observed input range. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$residual=y_1−model(x_1)$

Worked examples

Try itA fitted model is ŷ = 2x + 9. At x = 27, the observed output is 59. What is the residual?Show answer

Prediction=2(27)+9=63. Residual=observed−predicted=59−63=−4. The point lies four output units below the fitted line. Therefore the requested answer is -4.

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