Skip to main content
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 106
Chapter 3, Problem 106

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

Verified step by step guidance
1
Start by graphing the standard cubic function f(x) = x³. This is the parent function, which has a characteristic S-shaped curve passing through the origin (0, 0). The graph is symmetric about the origin, increasing to the right and decreasing to the left.
Identify the transformations applied to the parent function. The given function is h(x) = (1/2)(x − 2)³ − 1. Notice the transformations: (x − 2)³ represents a horizontal shift 2 units to the right, (1/2) scales the graph vertically by a factor of 1/2, and −1 shifts the graph downward by 1 unit.
Apply the horizontal shift. To shift the graph of f(x) = x³ to the right by 2 units, replace x with (x − 2). This gives the intermediate function g(x) = (x − 2)³.
Apply the vertical scaling. Multiply the intermediate function g(x) = (x − 2)³ by 1/2 to compress the graph vertically. This results in the function h₁(x) = (1/2)(x − 2)³.
Apply the vertical shift. Subtract 1 from h₁(x) = (1/2)(x − 2)³ to shift the graph downward by 1 unit. The final function is h(x) = (1/2)(x − 2)³ − 1. Plot this graph by applying all transformations step by step to the parent function.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
10m
Was this helpful?

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 standard cubic function, f(x) = x³, has a characteristic 'S' shape and passes through the origin. Understanding its basic shape and properties, such as its inflection point and end behavior, is crucial for graphing transformations.
Recommended video:
4:56
Function Composition

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For example, the function h(x) = (1/2)(x − 2)³ – 1 represents a horizontal shift to the right by 2 units, a vertical stretch by a factor of 1/2, and a downward shift by 1 unit. Mastery of these transformations allows for the accurate graphing of modified functions based on their parent functions.
Recommended video:
4:22
Domain & Range of Transformed Functions

Graphing Techniques

Graphing techniques involve plotting points, identifying key features such as intercepts and turning points, and understanding the overall shape of the graph. For cubic functions, it is important to recognize the symmetry and behavior at infinity. Using transformations, one can apply these techniques to accurately represent the modified function h(x) based on the standard cubic function.
Recommended video:
Guided course
02:16
Graphs and Coordinates - Example
Related Practice
Textbook Question

Begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = ∛x+2

Textbook Question

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

Textbook Question

In Exercises 107–108, write the standard form of the equation of the circle with the given center and radius. Center (-2. 4), r = 6

1
views
Textbook Question

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

8
views
Textbook Question

Begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = ∛(x-2)

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.

2
views