/*! 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 16 Sketch the graph of \(g(x)=|x+3|... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 16

Sketch the graph of \(g(x)=|x+3|-4\) by first sketching \(h(x)=|x|\) and then translating.

Short Answer

Expert verified
Translate \(|x|\) left 3 units and down 4 units.

Step by step solution

01

Understand the basic graph

The function to start with is the absolute value function, which is \(h(x) = |x|\). The graph of \(h(x) = |x|\) is a V-shaped graph that opens upwards with the vertex at the origin (0,0). Identify this shape and the position.
02

Translate horizontally

The expression inside the absolute value function in \(g(x) = |x+3| - 4\) is \(x+3\). This means we need to perform a horizontal translation of the \(h(x) = |x|\) graph. Replace \(x\) with \(x+3\) which shifts the graph 3 units to the left. Now the vertex of \(|x+3|\) is at (-3,0).
03

Translate vertically

Next, adjust for the '-4' outside the absolute value to complete the translation. This shifts the graph down by 4 units. The vertex of the graph \(g(x) = |x+3| - 4\) is now at (-3,-4).
04

Sketch the final graph

Draw the translated V-shaped graph with the new vertex at (-3, -4). The arms of the 'V' rise upwards at 45-degree angles from the vertex.

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.

Function Transformations
Function transformations help us understand how changes to a function's equation affect its graph. They can drastically change the position, shape, and orientation of a graph on a coordinate plane. Here are some fundamental transformations:
  • Translation: This moves the entire graph horizontally or vertically without altering its shape or orientation.
  • Reflection: This flips the graph over a line, such as the x-axis or y-axis, creating a mirror image.
  • Stretching/Shrinking: This alters the graph's shape in a specific direction, either making it wider or narrower.
In the exercise, we focus on translation, observing how these movements modify the graph of the absolute value function to obtain the desired graph of the transformed function.
Horizontal Translation
Horizontal translation involves shifting a graph left or right along the x-axis. When analyzing expressions like \(x + c\), the term \(c\) indicates the movement direction and magnitude:
  • If \(x + c\), move the graph \(|c|\) units left if \(c > 0\).
  • If \(x - c\), move the graph \(|c|\) units right if \(c > 0\).
In the given function \(|x + 3|\), the graph of \(h(x) = |x|\) shifts 3 units left.
Hence, the vertex initially at the origin (0,0) moves to (-3,0). This is a crucial step, as it sets the stage for further transformations. Remember to always check the signs to determine the correct direction of movement.
Vertical Translation
Vertical translation shifts a graph up or down along the y-axis. This movement is dictated by the constant term outside the function. In the exercise, \(|x+3| - 4\), changes occur vertically:
  • A positive constant, say \(d\), shifts the graph \(|d|\) units up.
  • A negative constant, such as \(-4\) in our case, shifts it \(|-4| = 4\) units down.
Following the horizontal shift, the graph's new vertex at (-3,0) descends 4 units to (-3,-4). This adjustment completes the transformation, aligning the graph accurately on the coordinate plane. Vertical translations are straightforward: look to the constant, then shift accordingly.
Graph Sketching
Once all transformations are understood, sketching the graph is a simple process. Begin with the function \(h(x) = |x|\), a V-shaped graph centered at the origin. Apply each transformation step-by-step:
  • First: Shift left 3 units from (0,0) to (-3,0).
  • Then: Shift down 4 units to arrive at the final vertex position of (-3,-4).
This produces a translated V-shape with its arms extending upward at 45-degree angles from the vertex. Be sure to maintain the V's symmetry during sketching. Finally, label your axes and vertex clearly for clarity. Understanding graph transformations allows for accurate sketching of any given function, providing a visual representation of its behavior.

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

Find a formula for \(f^{-1}(x)\) and then verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\) $$ f(x)=x^{5 / 2}, x \geq 0 $$

Find the value of \(k\) such that the line \(k x-3 y=10\) (a) is parallel to the line \(y=2 x+4\); (b) is perpendicular to the line \(y=2 x+4\); (c) is perpendicular to the line \(2 x+3 y=6\).

Verify that $$ \frac{\pi}{4}=4 \tan ^{-1}\left(\frac{1}{5}\right)-\tan ^{-1}\left(\frac{1}{239}\right) $$ a result discovered by John Machin in 1706 and used by him to calculate the first 100 decimal places of \(\pi .\)

Show that the two circles \(x^{2}+y^{2}-4 x-2 y-11=0\) and \(x^{2}+y^{2}+20 x-12 y+72=0\) do not intersect. Hint: Find the distance between their centers.

In Problems 17-22, find the center and radius of the circle with the given equation. x^{2}+y^{2}-12 x+35=0
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Decision 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.