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AP Precalculus Section 4.1 Example: Graphing Parametric Equations (Parabola)
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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!
Graphing parametric equations for a parabola involves several steps:
1. **Parameterization**: Parametrize the parabola in terms of a parameter, typically denoted as \( t \). For a parabola, the parameterization might be in the form \( x = f(t) \) and \( y = g(t) \), where \( f(t) \) and \( g(t) \) are functions of \( t \).
2. **Range of \( t \)**: Determine the range of values for \( t \) that you want to plot. This range will determine the portion of the parabola that you'll graph.
3. **Plotting Points**: Choose values of \( t \) within the determined range and plug them into the parameterized equations to obtain corresponding \( x \) and \( y \) coordinates.
4. **Connect Points**: Once you have a set of \( x \) and \( y \) coordinates, plot them on a coordinate plane and connect them smoothly to form the parabolic curve.
5. **Optional**: Label key points or indicate the direction of the curve if necessary.
6. **Check for Symmetry**: Parabolas are symmetric about their axis. Make sure your graph reflects this symmetry.
7. **Adjustments**: Make any necessary adjustments to the scale or viewing window to ensure the entire parabolic curve is visible and well-represented on the graph.
By following these steps, you can accurately graph parametric equations for a parabola.
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
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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