AP Precalculus Section 3.1 Example: Solving Using a Periodic Function and Graph

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

Solving a periodic function typically involves finding its properties, such as amplitude, period, phase shift, and frequency. Here are the general steps to solve a periodic function:

1. **Identify the Type of Function:**
Determine whether the function is a trigonometric function (e.g., sine, cosine) or another type of periodic function.

2. **Find the Amplitude:**
If the function is trigonometric, identify the amplitude, which is the maximum distance the function reaches from its midline.

3. **Determine the Period:**
Find the period of the function, which is the length of one complete cycle. For trigonometric functions, this is often related to the coefficient of the variable inside the trigonometric function.

4. **Check for Phase Shift:**
Determine if there is a phase shift, which is a horizontal shift of the function along the x-axis. This is usually indicated by a term inside the function.

5. **Calculate Frequency:**
For trigonometric functions, frequency is related to the reciprocal of the period.

6. **Write the Equation:**
Use the gathered information to write the equation of the periodic function.

7. **Graph the Function:**
Optionally, you can graph the function to visualize its behavior.

8. **Solve Equations or Inequalities:**
If the problem involves solving equations or inequalities with the periodic function, apply standard algebraic techniques.

Remember that the specific steps may vary based on the type of periodic function you're dealing with. It's important to be familiar with the properties of the particular type of function you are working with. If you have a specific periodic function in mind, feel free to provide more details for a more tailored explanation.

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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