AP Precalculus Section 4.1 Example: Write Parametric Equations in Rectangular Form

NOTE: Unit 4 is not included in the official AP Precalculus exam, but is still included in the curriculum.

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

Writing parametric equations in rectangular form involves converting equations that are given in terms of a parameter (typically denoted as \( t \)) into equations in terms of \( x \) and \( y \).

Here are the steps:

1. **Parameterization**: Start with equations given in terms of a parameter, usually denoted as \( t \). For example, you might have \( x = f(t) \) and \( y = g(t) \), where \( f(t) \) and \( g(t) \) are functions of \( t \).

2. **Isolate \( t \)**: If possible, solve one of the equations for \( t \). This allows you to express \( t \) in terms of one of the variables, usually \( x \) or \( y \). If both equations are solved for \( t \), you can use either.

3. **Substitute for \( t \)**: Substitute the expression for \( t \) obtained in step 2 into the other equation. This will give you an equation in terms of only one variable, either \( x \) or \( y \).

4. **Simplify**: Simplify the equation obtained in step 3 to express it solely in terms of one variable, either \( x \) or \( y \). This will be the rectangular form of the parametric equations.

5. **Check for Constraints**: Ensure that any constraints on the variable \( t \), such as the range of \( t \), are properly translated into constraints on the variables \( x \) and \( y \) in the rectangular form.

By following these steps, you can convert parametric equations into rectangular form.

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
4.1 Parametric Functions
4.2 Parametric Functions Modeling Planar Motion
4.3 Parametric Functions and Rates of Change
4.4 Parametrically Defined Circles and Lines
4.5 Implicitly Defined Functions
4.6 Conic Sections
4.7 Parametrization of Implicitly Defined Functions
4.8 Vectors
4.9 Vector-Valued Functions
4.10 Matrices
4.11 The Inverse and Determinant of a Matrix
4.12 Linear Transformations and Matrices
4.13 Matrices as Functions
4.14 Matrices Modeling Contexts

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

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