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
4:24 minutes
Problem 51b
Textbook Question
Textbook QuestionConnecting Graphs with Equations Find a quadratic function f having the graph shown. (Hint: See the Note following Example 3.)
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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) = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the general shape and properties of parabolas is essential for connecting their equations to their graphical representations.
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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 quadratic function in vertex form, f(x) = a(x - h)² + k, the vertex is located at the point (h, k). In the given graph, the vertex is at (3, -9), which indicates the minimum value of the function and is crucial for determining the function's equation.
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Standard Form of a Quadratic Equation
The standard form of a quadratic equation is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form allows for easy identification of the vertex and the direction in which the parabola opens. To find the specific quadratic function that matches the given graph, one can substitute the vertex coordinates into this form and determine the value of 'a' based on additional points on the graph.
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