Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 101

Chapter 3, Problem 101

Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. h(x) = x³/2

Verified step by step guidance

Start by identifying the standard cubic function, f(x) = x³. This is a basic cubic function with a graph that passes through the origin (0, 0) and has a characteristic S-shape, increasing to the right and decreasing to the left.

Recognize that the given function, h(x) = x³/2, is a transformation of the standard cubic function. Specifically, dividing the cubic term by 2 represents a vertical compression of the graph by a factor of 1/2.

To apply the vertical compression, take each y-coordinate of the points on the graph of f(x) = x³ and multiply it by 1/2. For example, if a point on f(x) is (1, 1), it becomes (1, 1/2) on h(x). Similarly, if a point is (-1, -1), it becomes (-1, -1/2).

Plot the transformed points on the graph and sketch the new curve. The overall shape of the graph remains the same (an S-shape), but it is vertically compressed, making it less steep compared to the original cubic function.

Label the graph of h(x) = x³/2 clearly, and ensure that the key points and the general behavior of the function are accurately represented.

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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 standard cubic function, f(x) = x³, has a characteristic S-shaped curve that passes through the origin and extends infinitely in both directions. Understanding the basic shape and properties of cubic functions is essential for analyzing transformations.

Graph Transformations

Graph transformations involve altering the position or shape of a function's graph through various operations, such as translations, reflections, stretches, and compressions. For example, the function h(x) = x³/2 represents a vertical compression of the standard cubic function f(x) = x³. Recognizing how these transformations affect the graph is crucial for accurately sketching the new function.

Vertical Compression

Vertical compression occurs when a function's output values are multiplied by a factor between 0 and 1, resulting in a graph that is 'squished' towards the x-axis. In the case of h(x) = x³/2, the factor of 1/2 compresses the graph of f(x) = x³ vertically, making it less steep. This concept is vital for understanding how the shape of the graph changes in response to the transformation applied.

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