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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 104
Chapter 3, Problem 104

Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. r(x) = (x − 2)³ +1

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Start by recalling the standard cubic function, f(x) = x³. This is a basic cubic graph that passes through the origin (0, 0), is symmetric about the origin, and has the general shape of an S-curve. The graph decreases for x < 0 and increases for x > 0.
Identify the transformations applied to the standard cubic function to obtain r(x) = (x − 2)³ + 1. The term (x − 2)³ represents a horizontal shift to the right by 2 units, and the +1 represents a vertical shift upward by 1 unit.
To graph r(x), first shift the graph of f(x) = x³ to the right by 2 units. This means that every point (a, b) on the graph of f(x) = x³ will move to (a + 2, b).
Next, shift the graph upward by 1 unit. This means that every point (a + 2, b) from the previous step will move to (a + 2, b + 1).
Finally, plot the transformed graph of r(x) = (x − 2)³ + 1. The new graph will still have the general S-curve shape of the cubic function, but its inflection point will now be at (2, 1) instead of the origin (0, 0).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cubic Functions

A cubic function is a polynomial function of degree three, typically expressed in the form f(x) = ax³ + bx² + cx + d. The graph of a standard cubic function, such as f(x) = x³, has a characteristic 'S' shape and passes through the origin. Understanding the basic shape and properties of cubic functions is essential for analyzing transformations.
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Function Composition

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, the function r(x) = (x − 2)³ + 1 represents a horizontal shift to the right by 2 units and a vertical shift upward by 1 unit from the standard cubic function. Mastery of these transformations allows for the accurate graphing of modified functions.
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Intro to Transformations

Function Notation and Evaluation

Function notation, such as f(x) or r(x), is a way to represent a function and its output for a given input x. Evaluating a function involves substituting a specific value for x to find the corresponding output. Understanding how to manipulate and evaluate functions is crucial for applying transformations and analyzing their effects on the graph.
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Evaluating Composed Functions
Related Practice
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