AP Precalculus Section 1.13 Example: Is the Table Linear, Quadratic, or Cubic?

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

To determine if a table of values represents a linear, quadratic, or cubic function using equal length input value intervals, follow these steps:

1. **Identify the Input Value Intervals:**
Ensure that the difference between consecutive input values (x-values) in your table is the same. For instance, if you have \(x\) values of 1, 3, 5, 7, the interval is 2 (difference between each pair of consecutive values).

2. **Calculate the Differences in Output Values (y-values):**
Once you've confirmed equal intervals for input values, calculate the differences between consecutive output values (y-values).

3. **Linear Function:**
- If the differences between consecutive output values are constant and the second differences are zero, the function is linear. For instance, if the differences between y-values are constant (e.g., 2, 4, 6, 8) and the second differences are zero, it indicates a linear relationship.

4. **Quadratic Function:**
- If the differences between consecutive output values are not constant but the differences between these differences (the second differences) are constant, the function is quadratic. For example, if the differences between y-values are not constant but the second differences are constant (e.g., 2, 6, 12, 20), it suggests a quadratic relationship.

5. **Cubic Function:**
- If the differences between consecutive output values are not constant, and neither are the second differences, but the differences between these second differences (the third differences) are constant, the function is cubic. For instance, if the differences between y-values are not constant, and the second differences are not constant, but the third differences are (e.g., 2, 12, 30, 56), it indicates a cubic relationship.

By analyzing the differences between consecutive output values, second differences, and, if necessary, third differences, you can identify whether the table of values represents a linear, quadratic, or cubic function based on equal-length intervals of input values.

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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