/*! 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 114 Begin by graphing the cube root ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 114

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

Short Answer

Expert verified
The graph of function \(r(x)\) starts from the point (2,2) and continues to rise at a slower rate than \(f(x)\) due to being vertically compressed by a factor of 1/2.

Step by step solution

01

Graph the cube root function

Begin by graphing the cube root function \(f(x) = \sqrt[3]{x}\). This function is a fairly simple function to graph as it increases and decreases at a steady rate. The graph will start from negative infinity, passing through the origin (0,0) and then rise to positive infinity.
02

Apply the horizontal shift

The graph will be horizontally shifted by two units to the right in comparison with the graph of \(f(x)\). This means that every point on the original \(f(x)\) graph will be moved two units to the right. Thus \(f(x) = \sqrt[3]{x}\) becomes \(f(x) = \sqrt[3]{x-2}\). The graph starts from the point (2,0) now after this shift.
03

Apply the vertical compression transformation

The graph would be vertically compressed by a factor of 1/2. This effectively reduces the vertical y-values of each point on the curve of the graph by half while leaving the x-values unchanged. So, \(f(x) = \sqrt[3]{x-2}\) turns into \(f(x) = \frac{1}{2}\sqrt[3]{x-2}\).
04

Apply the vertical shift

Finally, the function will be vertically shifted up by two units which means every point on the graph of \(f(x) = \frac{1}{2} \sqrt[3]{x-2}\) is moved two units upwards, resulting in the graph of \(r(x) = \frac{1}{2} \sqrt[3]{x-2} + 2\). The graph now correctly represents \(r(x)\).

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

(For assistance with this exercise, refer to the discussion of piecewise functions beginning on page 234 , as well as to Exercises \(79-80 .\) ) Group members who have cellphone plans should describe the total monthly cost of the plan as follows: \(\$ ________\) per month buys ____________ minutes. Additional time costs $\$$ __________per minute.

use a graphing utility to graph each circle whose equation is given. $$ (y+1)^{2}=36-(x-3)^{2} $$

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{aligned} (x-2)^{2}+(y+3)^{2} &=4 \\ y &=x-3 \end{aligned} $$

complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}+12 x-6 y-4=0 $$

graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{aligned} x^{2}+y^{2} &=16 \\ x-y &=4 \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

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.