AP Precalculus Section 4.6 Example: Graph an Ellipse in General Form (Conic Sections)

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!

To graph an ellipse in general form, follow these steps:

1. **Understand the General Form:** The general form of an ellipse equation is:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
where \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) are constants.

2. **Identify the Center:** Complete the square to rewrite the equation in standard form. The completed square form reveals the center of the ellipse.

3. **Determine Major and Minor Axes:** From the completed square form, identify the lengths of the major and minor axes. These lengths will be related to the coefficients of \(x^2\) and \(y^2\) terms.

4. **Plot the Center:** Plot the center of the ellipse on the coordinate plane.

5. **Plot the Axes:** Draw axes through the center of the ellipse corresponding to the major and minor axes.

6. **Plot the Ellipse:** Use the lengths of the major and minor axes to mark points along the axes. Connect these points with a smooth curve to sketch the ellipse.

7. **Optional: Determine Additional Points:** If needed, find and plot additional points to ensure accuracy.

8. **Label Axes and Center:** Label the axes with their respective lengths and label the center point.

9. **Finalize the Graph:** Add any necessary details to make the graph clear and accurate.

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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