AP Precalculus Section 3.1 Example: Finding a Value and Average Rate of Change in Periodic Functions

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

If you have a periodic function with a period of 6 and you want to find the value of the graph at \( x = 21 \), you can use the periodic nature of the function to simplify the calculation. Here are the steps:

1. **Identify the Period:**
Given that the period of the function is 6, it means that the function repeats every 6 units along the x-axis.

2. **Determine the Equivalent Position within One Period:**
Find the equivalent position of \( x = 21 \) within one period. You can do this by dividing \( 21 \) by the period \( 6 \):
\[ 21 \div 6 = 3 \, \text{(with a remainder of 3)} \]

This means that \( x = 21 \) is equivalent to \( x = 3 \) within one period.

3. **Evaluate the Function at the Equivalent Position:**
Evaluate the function at the equivalent position \( x = 3 \). This will give you the value of the graph at \( x = 21 \).

If the function is represented by \( f(x) \), then find \( f(3) \).

For example, if the function is a sine function:
\[ f(x) = \sin\left(\frac{2\pi}{6}x\right) \]
Plug in \( x = 3 \):
\[ f(3) = \sin\left(\frac{2\pi}{6} \times 3\right) \]

4. **Calculate the Value:**
Calculate the value of the function at \( x = 3 \). In this example:
\[ f(3) = \sin\left(\frac{\pi}{2}\right) \]

Evaluate this expression to get the numerical value.

By following these steps, you can find the value of a periodic function with a period of 6 at a specific point, such as \( x = 21 \). Adjust the function representation based on the specific periodic function you are working with.

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:

/ nickperich

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

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