Textbook Question
Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
As ______
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 11
Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)2+5
Verified step by step guidance1
Identify the form of the quadratic function. The given function is in vertex form: \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
Rewrite the function to match the vertex form template. The function is \(f(x) = -2(x + 1)^2 + 5\), which can be seen as \(f(x) = -2(x - (-1))^2 + 5\).
From the expression, determine the vertex coordinates by comparing to \(f(x) = a(x - h)^2 + k\). Here, \(h = -1\) and \(k = 5\).
Therefore, the vertex of the parabola is at the point \((-1, 5)\).
Remember that the coefficient \(a = -2\) indicates the parabola opens downward, but this does not affect the vertex coordinates.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertex Form of a Quadratic Function
The vertex form of a quadratic function is expressed as f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. This form makes it easy to identify the vertex directly from the equation without further calculations.
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Vertex Form
Effect of Parameters a, h, and k on the Parabola
In the vertex form, 'a' determines the parabola's direction and width: if a is negative, the parabola opens downward; if positive, upward. The values 'h' and 'k' shift the parabola horizontally and vertically, respectively, locating the vertex at (h, k).
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Finding the Vertex Coordinates
To find the vertex coordinates from a quadratic in vertex form, identify h and k from the expression f(x) = a(x - h)^2 + k. Note the sign inside the parentheses is opposite to the vertex's x-coordinate. The vertex is then (h, k), which gives the maximum or minimum point of the parabola.
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02:16Graphs and Coordinates - Example
Related Practice
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