/*! 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 57 Graph both functions in the same... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 57

Graph both functions in the same viewing window and describe how \(g\) is a transformation of $f . $$f(x)=x^{2}, g(x)=2 x^{2}-3$$

Short Answer

Expert verified
g(x) is a vertical stretch by a factor of 2 and a vertical shift downward by 3 units of f(x).

Step by step solution

01

- Understand the Functions

First, identify the two functions given: Function 1: \(f(x) = x^2\)Function 2: \(g(x) = 2x^2 - 3\)
02

- Graph the Parent Function

Graph the parent function \(f(x) = x^2\). This is a basic parabola that opens upwards with its vertex at the origin (0,0).
03

- Identify Transformations for g(x)

Compare \(g(x) = 2x^2 - 3\) to \(f(x) = x^2\). Notice there are two transformations:1. Vertical stretch by a factor of 2 (the coefficient 2).2. Vertical shift downward by 3 units (the constant term -3).
04

- Graph g(x)

Apply the transformations to \(f(x)\) to graph \(g(x)\):1. Stretch the graph of \(f(x)\) vertically by multiplying the y-values by 2. Each point \((x, y)\) on \(f(x)\) moves to \((x, 2y)\).2. Shift the entire graph downward by 3 units by subtracting 3 from each y-value of the transformed function.
05

- Compare Both Functions

Observe the differences:1. \(g(x)\) is narrower compared to \(f(x)\) because of the vertical stretch by a factor of 2.2. \(g(x)\) is shifted downward by 3 units. This moves the vertex from the origin (0,0) to the point (0,-3).
06

- Confirm with a Graphing Tool

Use a graphing calculator or software to plot \(f(x) = x^2\) and \(g(x) = 2x^2 - 3\) in the same viewing window. Verify the transformations: the stretching and the downward shift.

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 Functions
Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the inputs (x-values) and the outputs (y-values) of a function. With the function \(f(x) = x^2\), you get a parabola that opens upwards. Each point on the graph can be found by substituting an x-value into the function to get the corresponding y-value. For example, if \(x=2\), then \(f(2) = 2^2 = 4\), so the point (2, 4) is on the graph.

To graph functions, follow these steps:
  • Identify the function's formula.
  • Choose a range of x-values.
  • Calculate the corresponding y-values.
  • Plot the points on the coordinate plane.
  • Connect the points to see the shape of the graph.
This provides a clear, visual way to understand how the function behaves and helps identify any transformations applied to the parent function.
Vertical Stretch
A vertical stretch changes the steepness of the graph by multiplying the y-values by a constant factor, making the graph narrower or wider. For the function \(g(x) = 2x^2 - 3\), compared to \(f(x) = x^2\), the y-values are multiplied by 2. This means for any given x-value, you now have double the y-value of the original function. For instance, if \(f(1) = 1\), then \(g(1) = 2 \cdot 1^2 - 3 = 2 - 3 = -1\). The effect is a narrower curve because all points are stretched away from the x-axis.

Steps to apply a vertical stretch:
  • Identify the stretch factor. In \(g(x) = 2x^2 - 3\), it's 2.
  • Multiply each y-value of the parent function by this factor.
This transformation helps in modifying the sharpness of the graph without altering its general direction.
Vertical Shift
A vertical shift moves the entire graph up or down by a certain number of units. For the function \(g(x) = 2x^2 - 3\), the graph of \(2x^2\) is shifted down by 3 units because of the -3 term. This means every point on the function \(2x^2\) is moved down by 3 units. For example, if initially \(2x^2 = 8\) at some \(x\), then \(g(x) = 2x^2 - 3\) will be 5 (since \(8 - 3 = 5\)).

Steps to apply a vertical shift:
  • Identify the shift direction and magnitude. For \(g(x)\), it's 3 units downward.
  • Subtract (or add for upward shifts) this number from each y-value of the graph.
This transformation helps shift the entire graph without changing its shape.

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 the signs of the six trigonometric function values for the given angles. $$91^{\circ}$$

Find the function value. Round to four decimal places. $$\sin 3824^{\circ}$$

Use a calculator in Exercises \(81-84,\) but do not use the trigonometric function keys. Given that \(\sin 41^{\circ}=0.6561\) \(\cos 41^{\circ}=0.7547\) \(\tan 41^{\circ}=0.8693\) find the trigonometric function values for \(319^{\circ} .\)

$$\text {Graph each of the following.}$$ $$f(x)=|x| \cos x$$

Given that \(\sec \beta=1.5304,\) find \(\sin \left(90^{\circ}-\beta\right)\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

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.