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
2. Graphs of Equations
Two-Variable Equations
0:55 minutes
Problem 25
Textbook Question
Textbook QuestionDetermine whether each equation defines y as a function of x. y = 6 -x^2
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Definition
A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for a relation to be a function, no two ordered pairs can have the same first element with different second elements. This concept is crucial for determining if an equation defines y as a function of x.
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Vertical Line Test
The vertical line test is a visual way to determine if a curve is a graph of a function. If any vertical line intersects the graph at more than one point, then the graph does not represent a function. This test is particularly useful for analyzing the graphical representation of equations to confirm their functional nature.
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Quadratic Functions
A quadratic function is a type of polynomial function represented by the equation y = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, which can open upwards or downwards. Understanding the properties of quadratic functions helps in analyzing whether they define y as a function of x, particularly in terms of their symmetry and vertex.
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