/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Graph each function. \(y=\frac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 23

Graph each function. \(y=\frac{1}{2} \sqrt[3]{x-1}+3\)

Short Answer

Expert verified
The graph of the function \(y=\frac{1}{2} \sqrt[3]{x-1}+3\) is a cubic root graph, vertically stretched by a factor of 0.5, shifted 1 unit to the right and 3 units up. The domain and range are all real numbers, with x-intercept at \(x=1\) and y-intercept at \(y=\frac{5}{2}\).

Step by step solution

01

Determine the domain and range

The domain of the function is all real numbers since the cube root is defined for all real numbers. Therefore, Domain: \(-\infty < x < \infty\). The output of the cube root function is also all real numbers. With an additional vertical shift of 3 units up, the range will be all values greater than 3. Therefore, Range: \(-\infty < y < \infty\)
02

Find the x-intercept and y-intercept

To find the x-intercept, set \(y=0\) and solve for \(x\). This gives \(x=1\). For the y-intercept, set \(x=0\) and solve for \(y\). We get \(y=3 + \frac{1}{2} \sqrt[3]{-1} = 3 - \frac{1}{2} = \frac{5}{2}\)
03

Sketch the cubic root using transformations

Begin with the basic cubic root function \(y = \sqrt[3]{x}\). Then perform the transformations step by step. By the form \(y=a \sqrt[3]{b(x-h)}+k\), we can see that we have factor of 0.5 (a = 0.5) which is a vertical stretch by 0.5, a horizontal shift to the right by 1 unit (h = 1), and a vertical shift up by 3 units (k = 3). Make these transformations on the basic cubic root function to create the graph.
04

Plot the graph

Using information from Steps 1-3, plot the cubic root function with the transformations. Include the intercepts on your graph.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range
When working with functions, understanding the domain and range is essential.
The **domain** of a function refers to all the possible input values (or x-values) for which the function is defined.
In the case of the function \(y = \frac{1}{2} \sqrt[3]{x-1} + 3\), the cube root is defined for all real numbers.
  • This means the domain is all real numbers: \(-\infty < x < \infty\).

The **range** of a function refers to all the possible output values (or y-values) that the function can produce.
Since the cube root function itself can produce all real numbers, and the graph has been shifted vertically by 3 units, the range similarly includes all real numbers.
  • Thus, the range is also all real numbers: \(-\infty < y < \infty\).

In summary, for this cubic function, both domain and range span the entire set of real numbers.
Intercepts
Finding intercepts is a vital step in understanding the graph of a function.
**Intercepts** are the points where the graph crosses the axes of the coordinate plane.
For the x-intercept, you find where the graph crosses the x-axis. This occurs when the output or y-value is 0.
  • For our function \(y = \frac{1}{2} \sqrt[3]{x-1} + 3\), setting \(y = 0\) gives us \(x = 1\).

Whereas for the y-intercept, you determine where the graph intersects the y-axis. This happens when the input or x-value is 0.
  • Plugging \(x = 0\) into the function, the y-value is \( y= 3 - \frac{1}{2} = \frac{5}{2}\).

These intercepts, \(x = 1\) and \(y = \frac{5}{2}\), serve as crucial reference points when plotting the graph.
Transformations
Transformations play a key role in shifting and stretching the basic graph.
It allows you to shape the function effectively, providing a better understanding of its movement.
The given function is \(y = \frac{1}{2} \sqrt[3]{x-1} + 3\), and the transformations applied here can be divided into three types:
  • **Vertical Stretch**: The coefficient \(\frac{1}{2}\) in front applies a vertical stretch by a factor of 0.5.
    This makes the graph less steep than the parent function.
  • **Horizontal Shift**: The \(x-1\) inside the cube root shifts the graph 1 unit to the right on the x-axis.
  • **Vertical Shift**: Finally, we add 3 to the output, which shifts the entire graph up by 3 units.

By conducting these transformations step-by-step on the basic cube root graph \(y = \sqrt[3]{x}\), you can accurately obtain the desired plot.
This careful approach helps visualize how a function changes structurally.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{64 x-128}-3\)

What is the inverse of \(y=x^{2}-2 x+1 ?\) Is the inverse a function? Explain.

Explain the effect that \(a\) has on the graph of \(y=a \sqrt{x}\)

Graph. Find the domain and the range of each function. \(y=4 \sqrt[3]{x-2}+1\)

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. Find the domain and the range of each function. \(y=\sqrt{3 x-5}+6\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.