AP Precalculus Section 2.6 Example: Residual Plot (Example 2)

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

Residual plots are graphs that display the residuals, which are the differences between observed data values and the corresponding values predicted by a model. These plots help assess the goodness of fit of a model and detect patterns or trends in the residuals.

For a linear model:
- In a residual plot for a linear model, you'd ideally want to see a random scattering of points around the horizontal axis without any particular pattern.
- If the residuals form a random pattern around zero, it suggests that the linear model is appropriate for the data.
- Patterns in the residual plot, like a curve, a systematic increase or decrease in residuals as the predicted values change, or any distinct shape, indicate that the linear model might not be the best fit.

How to use residual plots to determine linearity:
1. **Random Scatter:** A good linear fit shows residuals scattered randomly around the horizontal axis.
2. **Patterns:** Look for any systematic patterns in the residuals. If you observe a pattern or trend (like a curve), it indicates that the linear model might not accurately capture the relationship between variables.
3. **Homoscedasticity:** Ensure that the spread of residuals remains consistent across all predicted values. Uneven or varying spreads might indicate non-linearity.

In summary, by examining the distribution and pattern of residuals in a residual plot, you can assess whether a linear model adequately represents the relationship between the variables. If the plot shows randomness and no discernible pattern, it suggests that a linear model is appropriate. Patterns or non-random behavior may indicate the need for a different model or transformations to better capture the data's nature.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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