Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 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

Verified step by step guidance

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.

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.

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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Related Practice

Textbook Question

Solve and graph the solution set on a number line: 3|2x-1| ≥ 21

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) = (1/2)(x − 2)³ – 1

Textbook Question

Exercises 103–105 will help you prepare for the material covered in the next section. Let (x1, y₁) = (7, 2) and (x2, y2) = (1, −1). Find √[(x2 − x1)² + (y2 − y₁)²]. Express the - answer in simplified radical form.

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Textbook Question

In Exercises 105–106, find the midpoint of each line segment with the given endpoints. (2, 6) and (-12, 4)

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Textbook Question

Exercises 103–105 will help you prepare for the material covered in the next section. Solve by completing the square: y² – 6y — 4 = 0.

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Textbook Question

Exercises 103–105 will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center (1, -1) and radius 1.

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