/*! 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 7 Graph each pair of equations on ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 7

Graph each pair of equations on one set of axes. $$y=|x| \text { and } y=|x-3|$$

Short Answer

Expert verified
The graph of \(y = |x|\) is a 'V' shape with vertex at (0,0), and \(y = |x - 3|\) is a 'V' shape with vertex at (3,0).

Step by step solution

01

Understand the Equations

The first equation is given as \(y = |x|\) which represents an absolute value function. This function creates a 'V' shape with its vertex at the origin (0,0). The second equation is given as \(y = |x - 3|\), which is also an absolute value function but shifted to the right by 3 units.
02

Identify Key Points for y = |x|

Identify key points for \(y = |x|\). When \(x = 0\), \(y = 0\). For \(x > 0\), \(y = x\) and for \(x < 0\), \(y = -x\). Key points include (0,0), (1,1), (2,2), (-1,1), (-2,2).
03

Graph y = |x|

Plot the points (0,0), (1,1), (2,2), (-1,1), (-2,2) on the graph. Connect them to form a 'V' shape with the vertex at (0,0).
04

Identify Key Points for y = |x - 3|

Identify key points for \(y = |x - 3|\). When \(x = 3\), \(y = 0\). For \(x > 3\), \(y = x - 3\) and for \(x < 3\), \(y = 3 - x\). Key points include (3,0), (4,1), (5,2), (2,1), (1,2).
05

Graph y = |x - 3|

Plot the points (3,0), (4,1), (5,2), (2,1), (1,2) on the graph. Connect them to form a 'V' shape with the vertex at (3,0).
06

Analyze the Graphs

Observe that \(y = |x|\) has its vertex at (0,0) and \(y = |x - 3|\) has its vertex at (3,0). Both graphs form a 'V' shape and are symmetric. The graph of \(y = |x - 3|\) is the same as \(y = |x|\) but shifted 3 units to the right.

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.

Graphing Absolute Value Functions
Absolute value functions create V-shaped graphs. When graphing the function, focus on key points and how they form this characteristic shape. For the equation, \(y = |x|\), it splits into two linear parts;
  • For \(x \geq 0\), \(y = x\)
  • For \(x < 0\), \(y = -x\)
These lines meet at the vertex, which is the lowest or highest point of the V-shape, located at the origin (0,0). You can plot points like (0,0), (1,1), (2,2), (-1,1), and (-2,2). Once plotted, they form the V shape with symmetry around the y-axis. Understanding how to plot these points accurately is essential for graphing absolute value functions.
Transformations of Functions
Transformations shift, stretch, or compress the graph. The equation \(y = |x - 3|\) shows a horizontal transformation of the base function \(y = |x|\). Specifically, it shifts the graph 3 units to the right. This happens because replacing \(x\) with \(x - 3\) means all x-values are effectively increased by 3. The key points now become:
  • (3,0) instead of (0,0)
  • (4,1) instead of (1,1)
  • (5,2) instead of (2,2)
  • (2,1) instead of (-1,1)
  • (1,2) instead of (-2,2)
The vertex, originally at (0,0), is now at (3,0). Transformations are crucial to understanding how basic functions change in different scenarios.
Vertex of Absolute Value Functions
The vertex is a key feature of absolute value functions. It represents the point where the two linear parts of the function meet. For the function \(y = |x|\), the vertex is at the origin (0,0). When transformations are applied, their new vertices can be found by looking at how the function is modified. In the equation \(y = |x - 3|\), the graph shifts horizontally. Consequently, the vertex moves from (0,0) to (3,0). Identifying the vertex helps in accurately drafting the V shape of the function, confirming the position and nature of the graph's minimal or maximal point.

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 the domain of each function. $$f(x)=\frac{x^{2}-4}{x+2}$$

In 2000 the percentage of 18 - to 29 -year-olds who used the Internet was \(72 \%\). In 2009 , that percentage had risen to \(92 \%\) a) Use the year as the \(x\) -coordinate and the percentage as the \(y\) -coordinate. Find the equation of the line that contains the data points. b) Use the equation in part (a) to estimate the percentage of Internet users in 2010 . c) Use the equation in part (a) to estimate the year in which the percentage of Internet users will reach \(100 \%\) d) Explain why a linear equation cannot be used for years after the year found in part(c).

Find the domain of each function given below. \(g(x)=\frac{2 x-6}{x^{2}-6 x+5} \quad\) (Hint: Factor the denominator)

Find the domain of each function given below. $$g(x)=|x+7|$$

Find the equilibrium point for the following demand and supply functions, where \(q\) is the quantity, in thousands of units, and \(x\) is the price per unit, in dollars. Demand: \(q=83-x\) Supply: \(\quad q=\frac{x^{2}}{576}-1.9\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.