Textbook Question
In Exercises 59-64, let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. g (f[h (1)])
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Ch. 2 - Functions and Graphs
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 3, Problem 62
In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x2. Then use transformations of this graph to graph the given function. r(x) = -(x + 1)2
Verified step by step guidance1
Start by recalling the graph of the standard quadratic function \(f(x) = x^2\). This is a parabola opening upwards with its vertex at the origin \((0,0)\).
Identify the given function \(r(x) = -(x + 1)^2\). Notice that it is a transformation of \(f(x) = x^2\).
Recognize the transformations: the expression \((x + 1)\) inside the squared term indicates a horizontal shift. Specifically, \(x + 1\) means the graph shifts 1 unit to the left.
The negative sign in front of the squared term, \(- (x + 1)^2\), reflects the graph over the x-axis, changing the parabola to open downwards instead of upwards.
Combine these transformations: start with the graph of \(f(x) = x^2\), shift it 1 unit left, then reflect it over the x-axis to get the graph of \(r(x) = -(x + 1)^2\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Quadratic Function
The standard quadratic function is f(x) = x², which graphs as a parabola opening upwards with its vertex at the origin (0,0). It serves as the base graph for understanding transformations applied to quadratic functions.
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Converting Standard Form to Vertex Form
Graph Transformations
Graph transformations involve shifting, reflecting, stretching, or compressing the base graph. For example, adding or subtracting inside the function shifts the graph horizontally, while multiplying by a negative reflects it across the x-axis.
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5:25Intro to Transformations
Reflection Across the x-axis
Multiplying a function by -1 reflects its graph across the x-axis, flipping it upside down. For r(x) = -(x + 1)², this means the parabola opens downward instead of upward.
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5:00Reflections of Functions
Related Practice
Textbook Question
Find a. (fog) (x) b. (go f) (x). f(x) = √x, g(x) = x − 1
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Textbook Question
In Exercises 59-66, a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. 8x – 4y – 12 =0
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Textbook Question
Use the vertical line test to identify graphs in which y is a function of x.
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Textbook Question
Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (1/2)(x − 1)²
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Textbook Question
Find c. (fog) (2) d. (go f) (2). f(x) = √x, g(x) = x − 1
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