/*! 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 86 Explain the difference between t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 86

Explain the difference between the graphs of \(f(x)=|x-2|-3\) and \(g(x)=|x-3|-2 .\)

Short Answer

Expert verified
The graph of \( f(x) = |x-2|-3 \) is shifted 2 units right and 3 units down. The graph of \( g(x) = |x-3|-2 \) is shifted 3 units right and 2 units down.

Step by step solution

01

- Understand the Parent Function

The parent function for both equations is the absolute value function, denoted as \( y = |x| \). This function forms a 'V' shape on the graph with its vertex at the origin.
02

- Identify Horizontal Shifts

For \( f(x) = |x-2|-3 \), the expression inside the absolute value is \( x-2 \). This indicates a horizontal shift 2 units to the right. For \( g(x) = |x-3|-2 \), the expression inside the absolute value is \( x-3 \), indicating a horizontal shift 3 units to the right.
03

- Identify Vertical Shifts

For \( f(x) = |x-2|-3 \), the \( -3 \) outside the absolute value indicates a vertical shift 3 units down. For \( g(x) = |x-3|-2 \), the \( -2 \) outside the absolute value indicates a vertical shift 2 units down.
04

- Compare the Final Graphs

After applying the shifts, the vertex of \( f(x) \) is at (2, -3) and for \( g(x) \) it is at (3, -2). The graphs are both 'V' shapes, but \( f(x) \) is shifted 2 units right and 3 units down, while \( g(x) \) is shifted 3 units right and 2 units down.

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.

Graph Transformations
Graph transformations are changes applied to the graph of a function, altering its position, shape, or size. For absolute value functions, these transformations can include horizontal shifts, vertical shifts, reflections, and stretching or compressing.
When transforming absolute value functions, start by observing the parent function. The parent function for an absolute value is represented as:
\[ y = |x| \]
This function has a distinct 'V' shape with its vertex at the origin (0, 0).
By applying transformations to this parent function, you can move the graph around and adjust its shape.
Horizontal Shift
A horizontal shift involves moving the graph left or right without altering its shape. To identify a horizontal shift in an absolute value function, look at the expression inside the absolute value symbols:
For instance, in the function:
\[ f(x) = |x-2| \]
The term \( x-2 \) indicates that the graph of the parent function \( y = |x| \) is shifted 2 units to the right.
This pattern holds for any function where you see \( x-a \) within the absolute value, where 'a' represents the number of units shifted to the right. If you see \( x+a \), the graph shifts 'a' units to the left.
In the provided exercise, the function \( f(x) = |x-2|-3 \) shifts 2 units right, while \( g(x) = |x-3|-2 \) shifts 3 units right.
Vertical Shift
A vertical shift moves the entire graph up or down. To identify a vertical shift, observe the constant term added or subtracted outside the absolute value function:
For example, in:
\[ f(x) = |x| - 3 \]
The \( -3 \) indicates a downward shift of 3 units.
Similarly, in
\[ g(x) = |x| + 2 \]
The \( +2 \) indicates an upward shift by 2 units.
In the exercise, the function \( f(x) = |x-2|-3 \) shifts 3 units down, while \( g(x) = |x-3|-2 \) shifts 2 units down.
Vertex
The vertex of an absolute value function is the point where the graph changes direction, forming a 'V' shape. It's crucial to determine as it helps in sketching the graph post-transformations.
For the parent function \( y = |x| \), the vertex is at (0, 0).
After applying horizontal and vertical shifts, the vertex moves to a new location:
  • For \( f(x) = |x-2| - 3 \), the vertex shifts to (2, -3).
  • For \( g(x) = |x-3| - 2 \), the vertex shifts to (3, -2).
By understanding the vertex shifts, you can graph absolute value functions accurately, ensuring they reflect the correct transformations.

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 if the function is even, odd, or neither. $$ m(x)=-4 x^{5}+2 x^{3}+x $$

A car traveling \(60 \mathrm{mph}(88 \mathrm{ft} / \mathrm{sec})\) undergoes a constant deceleration until it comes to rest approximately 9.09 sec later. The distance \(d(t)\) (in ft) that the car travels \(t\) seconds after the brakes are applied is given by \(d(t)=-4.84 t^{2}+88 t,\) where \(0 \leq t \leq 9.09 .\) (See Example 5) a. Find the difference quotient \(\frac{d(t+h)-d(t)}{h}\). Use the difference quotient to determine the average rate of speed on the following intervals for \(t\) : b. [0,2]\(\quad(\) Hint \(: t=0\) and \(h=2)\) c. [2,4]\(\quad(\) Hint \(: t=2\) and \(h=2)\) d. [4,6]\(\quad(\) Hint \(: t=4\) and \(h=2)\) e. [6,8]\(\quad(\) Hint \(: t=6\) and \(h=2)\)

The cost to buy tickets online for a dance show is \(\$ 60\) per ticket. a. Write a function that represents the cost \(C(x)\) (in \(\$$ ) for \)x\( tickets to the show. b. There is a sales tax of \)5.5 \%\( and a processing fee of \)\$ 8.00\( for a group of tickets. Write a function that represents the total cost \)T(a)\( for \)a\( dollars spent on tickets. c. Find \)(T \circ C)(x)\(. d. Find \)(T \circ C)(6)$ and interpret its meaning in the context of this problem.

Given \(f(x)=\sqrt{x+3}\) a. Find the difference quotient. b. Rationalize the numerator of the expression in part (a) and simplify. c. Evaluate the expression in part (b) for \(h=0\).

In computer programming the greatest integer function is sometimes called the "floor" function. Programmers also make use of the "ceiling" function which returns the smallest integer not less than \(x .\) For example: ceil( 3.1\()=4\). For Exercises \(115-116\), evaluate the floor and ceiling functions for the given value of \(x\). Floor \((x)\) is the greatest integer less than or equal to \(x\). Ceil \((x)\) is the smallest integer not less than \(x\). a. floor (2.8) b. floor (-3.1) c. floor(4) d. \(\operatorname{ceil}(2.8)\) e. \(\operatorname{ceil}(-3.1)\) f. ceil(4)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.