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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 97
Chapter 3, Problem 97

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

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Start by recalling the standard cubic function f(x) = x³. This function has a characteristic S-shaped curve, passing through the origin (0, 0), with symmetry about the origin. The graph increases to the right and decreases to the left.
Understand the transformation applied to f(x) = x³ to obtain g(x) = (x − 3)³. The term (x − 3) indicates a horizontal shift. Specifically, the graph of f(x) = x³ is shifted 3 units to the right.
To graph g(x) = (x − 3)³, take each key point on the graph of f(x) = x³ (e.g., (-1, -1), (0, 0), (1, 1), etc.) and shift it 3 units to the right. For example, the point (0, 0) on f(x) becomes (3, 0) on g(x).
Sketch the new graph by connecting the shifted points smoothly, maintaining the same S-shaped curve as the original cubic function. Ensure the graph still increases to the right and decreases to the left, with the inflection point now at (3, 0).
Label the graph of g(x) = (x − 3)³ clearly, and verify that the transformation has been applied correctly by checking a few additional points. For example, if x = 4, g(4) = (4 − 3)³ = 1³ = 1, so the point (4, 1) should be on the graph.

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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 cubic function has a characteristic 'S' shape and can have one or two turning points. Understanding the basic shape and behavior of the standard cubic function, f(x) = x³, 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 g(x) = (x - 3)³ represents a horizontal shift of the standard cubic function f(x) = x³ to the right by 3 units. Recognizing how these transformations affect the graph is crucial for accurately sketching the transformed function.
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Intro to Transformations

Function Notation and Evaluation

Function notation, such as f(x) or g(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 necessary for applying transformations and analyzing the resulting graphs.
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Evaluating Composed Functions
Related Practice
Textbook Question

Let f and g be defined by the following table: Find f(−1)−f(0)−[g(2)]2+f(−2)÷g(2)⋅g(−1).\(\sqrt{f(-1)-f(0)}\)-[g(2)]^2+f(-2)\(\div\) g(2)\(\cdot\) g(-1).

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Textbook Question

Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. h(x) = -x³

Textbook Question

In Exercises 95–96, let f and g be defined by the following table: Find |ƒ(1) − f(0)| − [g (1)]² +g(1) ÷ ƒ(−1) · g (2) .

Textbook Question

Which graphs in Exercises 96–99 represent functions that have inverse functions?

Textbook Question

Solve by completing the square: 2x² – 5x + 1 = 0.

Textbook Question

Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. g(x) = x³-3

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