/*! 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 22 Graph each equation .Let $x=-3,-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 22

Graph each equation .Let $x=-3,-2,-1,0,1,2, and 3. $$ y=-2|x| $$

Short Answer

Expert verified
When x=-3, -2, -1, 0, 1, 2, and 3 the corresponding y values are 6, 4, 2, 0, -2, -4, -6 respectively. Thus, graph of the function y=-2|x| is a 'V' shaped curve opening downwards centered on the origin (0,0), as all absolute value functions. But compared to the basic absolute function, y=|x|, this graph is stretched vertically by factor of 2 and reflected about x-axis because of the negative sign.

Step by step solution

01

Understand the Function

The function given is \(y=-2|x|\), where \(|x|\) represents absolute value of x and -2 is the coefficient. Absolute value makes all input values positive irrespective of their original sign.
02

Calculate the Value of y for Each x

Start by substituting each value of x in the function. For x=-3, -2, -1, 0, 1, 2, 3, the corresponding y values are found by calculating -2 multiplied by the absolute value of each x. This will give us values of y as 6, 4, 2, 0, -2, -4, -6 respectively.
03

Plot the Graph

Plot each pair of x and y on the graph. For any value of x, the corresponding y is the point on the y-axis. Join all these points to get the graph of the function.

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.

Understanding Absolute Value
The absolute value \( |x| \) refers to the distance of a number \( x \) from zero on the number line. It makes any negative number positive, as it measures only how far a number is from zero, removing any directional aspect. For example:
  • \( |3| = 3 \)
  • \( |-3| = 3 \)
When graphing functions involving absolute value, it helps to think about this transformation that "flips" negative inputs to their positive counterparts. This behavior is critical in our exercise as we handle the equation \(y = -2|x|\). Here, the \(|x|\) part assures that our inputs are positive or zero before further calculations.
Plotting Points on a Graph
Plotting points is crucial for visualizing functions. It involves placing points on a graph based on specified coordinates. In our exercise, after calculating \(y = -2|x|\) for given \(x\) values like -3, -2, -1, 0, 1, 2, and 3, we have specific \(y\) outputs like 6, 4, 2, 0, -2, -4, -6. Each \( (x, y) \) pair represents a point.For example, the point \( (1, -2) \) places the dot at 1 on the x-axis and -2 on the y-axis.
  • Make sure each value of \(x\) is plotted with its corresponding value of \(y\).
  • Ensure equal spacing on the axes to accurately portray each point's relative position.
By connecting these points sequentially, the overall graph's shape emerges, enabling interpretations of the function's nature.
Exploring Linear Transformations
Linear transformations involve changing a graph's position and shape without altering its general linearity. This function \( y = -2|x|\), incorporates a specific linear transformation due to the multiplication by -2.Here's how it works:
  • The coefficient -2 scales or stretches the graph vertically by a factor of 2. Every \(y\) value is doubled in magnitude.
  • The negative sign before 2 flips the graph across the x-axis. Thus, turning a typical V-shape of the absolute value into an upside-down V.
When handling transformations, visualizing these effects helps further understand a function's plot. Each transformation's essence lies in observing changes on the graph's structure, which in this exercise results in a reflected and vertically stretched absolute value function.

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 absolute value inequality. \(|x|>5\)

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(-4(x+2)>3 x+20\)

Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without discount passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.

Parts for an automobile repair cost \(\$ 175 .\) The mechanic charges \(\$ 34\) per hour. If you receive an estimate for at least \(\$ 226\) and at most \(\$ 294\) for fixing the car, what is the time interval that the mechanic will be working on the job?

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(1-\frac{x}{2}>4\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Calculus

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.