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
4. Polynomial Functions
Quadratic Functions
5:06 minutes
Problem 26
Textbook Question
Textbook QuestionGraph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. See Examples 1–4. ƒ(x) = (x - 5)^2 - 4
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Functions
A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the general shape and properties of parabolas is essential for analyzing their characteristics.
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Vertex of a Parabola
The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens downwards or upwards. For the function f(x) = (x - 5)^2 - 4, the vertex can be found directly from the vertex form of a quadratic equation, which is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex coordinates.
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Domain and Range
The domain of a function refers to all possible input values (x-values) that the function can accept, while the range refers to all possible output values (y-values) that the function can produce. For quadratic functions, the domain is typically all real numbers, while the range depends on the vertex and the direction the parabola opens, which can be determined from the vertex and the value of 'a'.
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