/*! 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 20 Use transformations of graphs to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 20

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=3-|x|, \quad y_{2}=3-|3 x|, \quad y_{3}=3-\left|\frac{1}{3} x\right|$$

Short Answer

Expert verified
The graphs of \(y_1\), \(y_2\), and \(y_3\) are transformations of the basic \(|x|\) graph involving reflection, stretching/compression, and vertical translation.

Step by step solution

01

Understand Basic Graph of |x|

Begin by recalling the basic graph of \( y = |x| \), which is a "V" shaped graph with the vertex at (0,0). The graph is symmetrical about the y-axis and has lines of slope 1 and -1.
02

Graph y1 = 3 - |x|

To sketch \( y_1 = 3 - |x| \), start with the graph of \( y = |x| \) and reflect it over the x-axis to get \( y = -|x| \), then translate this graph 3 units upwards. The vertex is now at (0,3), and the lines slope downwards at -1 and 1 from the vertex.
03

Graph y2 = 3 - |3x|

For \( y_2 = 3 - |3x| \), perform a horizontal compression by a factor of 3 on the basic \( |x| \) graph to make it \( |3x| \), resulting in a graph that is "steeper". Then, reflect this graph over the x-axis to invert it and translate it up by 3 units. The vertex still remains at (0,3) but the slope of the lines is now -3 and 3.
04

Graph y3 = 3 - |(1/3)x|

Sketch \( y_3 = 3 - |(1/3)x| \) by stretching the basic \( |x| \) graph horizontally by a factor of 3 to form \( |(1/3)x| \), which flattens the "V". Reflect it over the x-axis, and shift it up by 3 units. The vertex remains at (0,3), and the lines slope at -1/3 and 1/3.
05

Verify with Graphing Calculator

Enter the equations \( y_1 = 3 - |x| \), \( y_2 = 3 - |3x| \), and \( y_3 = 3 - |(1/3)x| \) in a graphing calculator. Check that the graphs match the expected shape and transformations described above, with appropriate slope and positions.

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.

Absolute Value Function
The absolute value function, denoted by \( |x| \), forms a fundamental building block for understanding graph transformations. This function produces a V-shaped graph, symmetrically reflecting across the y-axis, with its vertex at the origin (0,0). The absolute value captures the notion of distance from the origin on a number line. No matter what input you provide, the output remains non-negative.
  • The graph consists of two linear segments.
  • On the right side of the origin, it mirrors the equation \( y = x \).
  • On the left side, it mirrors \( y = -x \).
Understanding this simple function is crucial when manipulating it with various transformations. In the context of the exercise, these transformations include reflections, compressions, and vertical shifts.
Horizontal Compression
A horizontal compression involves squeezing the graph towards the y-axis. For transformations like \( y = |3x| \), the compression modifies the standard \( |x| \) graph. Specifically, each point on the graph moves closer to the y-axis by a factor related to the value multiplying the x-variable.
When compressing horizontally:
  • If \( y = |3x| \), it compresses the graph by a factor of 3. The graph appears steeper as it approaches the y-axis more sharply.
  • For \( y = |(1/3)x| \), it's a stretch by the factor of 3, making it less steep.
This transformation explains the changes in line slopes for equations \( y_2 \) and \( y_3 \) given in the exercise.
Vertical Shift
Vertical shifts involve moving the entire graph up or down along the y-axis. For any function \( y = |x| \, \pm \, b \), where \( b \) is a constant, the graph shifts vertically.
  • Addition of a positive number shifts the graph upwards.
  • A negative number shifts it downwards.
In the exercise, all graphs (\( y_1, y_2, y_3 \)) undergo a vertical upward shift by 3 units due to the "+3" included in each equation. Consequently, their vertices settle at the point (0,3) rather than the origin.
Graphing Calculator Verification
After sketching transformations manually, verifying your graph using a graphing calculator ensures precision. While drawing by hand offers foundational understanding, technology can confirm the accuracy of plotted lines and transformations. Here's how to leverage your calculator effectively:
  • Enter each equation into the calculator: \( y_1 = 3 - |x| \), \( y_2 = 3 - |3x| \), \( y_3 = 3 - |(1/3)x| \).
  • Ensure your calculator is set to the correct viewing window to capture all features of the graphs.
  • Check if the graph reflects your expected results 鈥� similar vertex positions with anticipated slopes.
Using a graphing calculator can solidify your understanding by providing a visual confirmation of your manual efforts.

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

Solve each application of openations and composition of functions. Dimensions of a Rectangle Suppose that the length of a rectangle is twice its width. Let \(x\) represent the width of the rectangle. (a) Write a formula for the perimeter \(P\) of the rectangle in terms of \(x\) alone. Then use \(P(x)\) notation to describe it as a function. What type of function is this? (b) Graph the function \(P\) as \(Y_{1}\) found in part (a) in the window \([0,10]\) by \([0,100]\). Locate the point for which \(x=4,\) and explain what \(x\) and \(y\) represent. (c) On the graph of \(P\), locate the point with \(x\) -value 4 . Then sketch a rectangle satisfying the conditions described carlier, and evaluate its perimeter if its width is this \(x\) -value. Use the standard perimeter formula. How does the result compare with the y-value shown on your screen? (d) On the graph of \(P\), find a point with an integer y-value. Interpret the \(x\) - and y-coordinates here.

Solve each problem. When a model kite was flown in crosswinds in tests, it attained speeds of 98 to 148 feet per second in winds of 16 to 26 feet per second. Using \(x\) as the variable in each case, write absolute value inequalities that correspond to these ranges.

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=\sqrt{x^{2}}$$

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=x-4\) must be less than 0.001 unit from 0.5.

Based on the ordered pairs seen in each table, make a conjecture about whether the finction \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r}x & f(x) \\\\-3 & -1 \\\\-2 & 0 \\\\-1 & 1 \\\0 & 2 \\\1 & 3 \\\2 & 4 \\\3 & 5\end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete 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.