/*! 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 49 49鈥�52 ? Graph the functions on... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 49

49鈥�52 ? Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-8,8]\) by \([-2,8]\) $$ \begin{array}{ll}{\text { (a) } y=\sqrt[4]{x}} & {\text { (b) } y=\sqrt[4]{x+5}} \\ {\text { (c) } y=2 \sqrt[4]{x+5}} & {\text { (d) } y=4+2 \sqrt[4]{x+5}}\end{array} $$

Short Answer

Expert verified
Each subsequent graph is a transformation of the previous, involving translations and stretches.

Step by step solution

01

Graph the function in part (a)

Graph the function \( y = \sqrt[4]{x} \) using the viewing rectangle \([-8, 8]\) by \([-2, 8]\) on the coordinate plane. This curve should start from the origin and increase slowly, showing a typical fourth root curve to the right.
02

Graph the function in part (b)

For the function \( y = \sqrt[4]{x+5} \), plot it in the same viewing rectangle. This graph is a horizontal translation 5 units to the left of the graph in part (a). The starting point now is at \( x = -5 \).
03

Graph the function in part (c)

Graph \( y = 2\sqrt[4]{x+5} \). This is a vertical stretch of the graph in part (b) by a factor of 2. The curve will rise quicker but starts from the same point \( x = -5 \).
04

Graph the function in part (d)

For \( y = 4 + 2 \sqrt[4]{x+5} \), graph this as a vertical translation of part (c). The entire graph of part (c) is shifted 4 units up, starting now from \( y = 4 \) at \( x = -5 \).

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.

Fourth root function
The fourth root function is a type of radical function represented by \( y = \sqrt[4]{x} \). This means that we are finding a number which, when multiplied by itself four times, equals \( x \). Fourth root functions generally exhibit a particular curve that begins at the origin and increases slowly if you consider positive values of \( x \).
  • For values of \( x \) greater than zero, \( y \) increases, showing a gentle upward curve.
  • The function is defined only for non-negative numbers since even roots of negative numbers are not real in a basic level context.
Fourth root graphs are typically mundane in the way they look like they crawl along the x-axis at first, gradually steepening as \( x \) increases. Understanding this characteristic is essential when you transform these functions algebraically or graphically.
Horizontal translation
Horizontal translation involves shifting a graph left or right along the x-axis. For the function \( y = \sqrt[4]{x+5} \), a horizontal translation is applied.
  • If \( h \) is positive, the function \( y = \sqrt[4]{x-h} \) shifts \( h \) units to the right.
  • In our case, \( y = \sqrt[4]{x+5} \) shifts the graph 5 units to the left.
You can imagine grasping the graph and simply moving it without altering its shape. This manipulation on the x-axis can significantly affect where the function starts (here at \( x = -5 \)), without altering its general shape. Understanding this allows for better control and prediction of changes in function behavior.
Vertical stretch
A vertical stretch happens when we multiply a function by a factor greater than one. This affects how quickly the values of \( y \) increase as \( x \) moves away from the origin.For the function \( y = 2\sqrt[4]{x+5} \), the original graph from \( y = \sqrt[4]{x+5} \) is vertically stretched by a factor of 2.
  • Instead of plodding neatly upwards, the curve rises more sharply.
  • This makes the function appear taller and steeper, amplifying its impact.
Visualizing vertical stretches helps in comparing and contrasting different functions on the same axes, allowing a clearer view of their relative changes and structures.
Vertical translation
Vertical translation is a shift of a graph up or down along the y-axis. This alters the starting height of the function without changing its shape.For \( y = 4 + 2\sqrt[4]{x+5} \), the graph from the previous step is lifted 4 units upwards.
  • The function no longer starts at \( y = 0 \), but at \( y = 4 \) when \( x = -5 \).
  • This move does not change the rate at which \( y \) increases as \( x \) does, yet repositions the entire curve.
Grasping vertical translation can particularly aid in real-world scenarios where initial values of a function need adjustments. It's like elevating a curve but maintaining its inherent nature.

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

61鈥�68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = {1} -{3}\sqrt{x}$$

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=(x-c)^{2}$$ (a) \(c=0,1,2,3 ;[-5,5]\) by \([-10,10]\) (b) \(c=0,-1,-2,-3 ; \quad[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

Volume of Water Between \(0^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C},\) the volume \(V\) (in cubic centimeters) of 1 \(\mathrm{kg}\) of water at a temperature \(T\) is given by the formula $$ V=999.87-0.06426 T+0.0085043 T^{2}-0.0000679 T^{3} $$ Find the temperature at which the volume of 1 \(\mathrm{kg}\) of water is a minimum.

\(45-50\) Express the function in the form \(f \circ g\) $$ F(x)=(x-9)^{5} $$

Determine whether the equation defines y as a function of x. (See Example 10.) $$ x=y^{4} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.