Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
3. Functions
Intro to Functions & Their Graphs
2:07 minutes
Problem 88
Textbook Question
Textbook QuestionDetermine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. y^3 = x + 4
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Symmetry in Graphs
Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations. For example, a graph is symmetric with respect to the x-axis if replacing y with -y yields the same equation, and symmetric with respect to the y-axis if replacing x with -x does. Origin symmetry occurs if both transformations hold true simultaneously.
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Graphs and Coordinates - Example
Testing for Symmetry
To test for symmetry, one can substitute values into the equation. For x-axis symmetry, replace y with -y and check if the equation remains valid. For y-axis symmetry, replace x with -x, and for origin symmetry, replace both x and y with their negatives. The results of these substitutions determine the type of symmetry present in the graph.
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Understanding Implicit Functions
The equation y^3 = x + 4 is an implicit function, meaning it defines y in terms of x without explicitly solving for y. Analyzing implicit functions often requires algebraic manipulation to explore their properties, including symmetry. Understanding how to work with implicit equations is crucial for determining their graphical characteristics.
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