/*! 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 111 Begin by graphing the cube root ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 111

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=\frac{1}{2} \sqrt[3]{x+2}$$

Short Answer

Expert verified
The graph of the function \(h(x)=\frac{1}{2} \sqrt[3]{x+2}\) is a transformation of the original cube root function \(f(x)=\sqrt[3]{x}\), where it is shifted two units to the left and vertically compressed by a factor of \(\frac{1}{2}\).

Step by step solution

01

Graph the basic function

Begin by graphing the basic cube root function: \(f(x)=\sqrt[3]{x}\). Remember, the cube root function starts from negative infinity, crosses the origin, and then continues to positive infinity.
02

Recognize the transformations

Identify the transformations present in the given function, \(h(x)=\frac{1}{2} \sqrt[3]{x+2}\). There are two transformations here: a horizontal shift (or translation) to the left by 2 units due to the '+2' inside the cube root, and a vertical dilation by a factor of \(\frac{1}{2}\) due to the multiplier outside the cube root.
03

Apply the horizontal shift

First, apply the horizontal shift by moving every point of the original function \(f(x)=\sqrt[3]{x}\) two units to the left. This is because of the '+2' inside the cube root.
04

Apply the vertical dilation

Then, apply the vertical dilation by reducing the y-value of every point by half. This is due to the multiplier of \(\frac{1}{2}\) outside the cube root.
05

Finalize the graph

Now, the transformed function \(h(x)=\frac{1}{2} \sqrt[3]{x+2}\) should be graphed. It should look like the original function \(f(x)=\sqrt[3]{x}\) that has been shifted two units to the left and vertically compressed by a factor of \(\frac{1}{2}\). The graph will start from negative infinity, reach a minimum at (-2,0), and then continue to positive infinity.

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!

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph is decreasing on \((-\infty, a)\) and increasing on \((a, \infty)\) so \(f(a)\) must be a relative maximum.

Show that the points \(A(1,1+d), B(3,3+d),\) and \(C(6,6+d)\) are collinear (lie along a straight line) by showing that the distance from \(A\) to \(B\) plus the distance from \(B\) to \(C\) equals the distance from \(A\) to \(C\).

Use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window. $$x^{2}+10 x+y^{2}-4 y-20=0$$

The toll to a bridge costs \(\$ 6.00 .\) Commuters who frequently use the bridge have the option of purchasing a monthly discount pass for \(\$ 30.00 .\) With the discount pass, the toll is reduced to \(\$ 4.00 .\) For how many bridge crossings per month will the cost without the discount pass be the same as the cost with the discount pass? What will be the monthly cost for each option?

Given an equation in \(x\) and \(y,\) how do you determine if its graph is symmetric with respect to the origin?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

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.