/*! 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 53 A function \(f\) is given, and t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 53

A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=x^{2} ;\) stretch vertically by a factor of \(2,\) shift downward 2 units, and shift 3 units to the right

Short Answer

Expert verified
The final transformed function is \(f(x) = 2(x - 3)^2 - 2\).

Step by step solution

01

Stretch Vertically

To stretch the graph vertically by a factor of 2, multiply the function by 2. The new function is given by \(g(x) = 2x^2\).
02

Shift Downward

To shift the graph downward by 2 units, subtract 2 from the function \(g(x)\). The function becomes \(h(x) = 2x^2 - 2\).
03

Shift to the Right

To shift the graph 3 units to the right, replace \(x\) with \((x - 3)\) in the function \(h(x)\). The final function is \(f(x) = 2(x - 3)^2 - 2\).

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.

Vertical Stretch
Vertical stretch is an important transformation that modifies the steepness of a graph, making it appear taller or narrower. For a given function, a vertical stretch is achieved by multiplying the entire function by a constant factor.
In our exercise, the function is originally \(f(x) = x^2\). To apply a vertical stretch by a factor of 2, we multiply the whole function by 2, resulting in \(g(x) = 2x^2\).
Key points to remember include:
  • The value of the factor determines how much taller the graph becomes. A factor greater than one results in a stretch.
  • If you multiply by less than one, it would be a compression, making the graph appear wider.
  • This transformation affects only the y-values of the function while the x-values remain unchanged.
By understanding vertical stretches, you can easily adjust and control the height or narrowness of different graphs, maintaining the same x-values throughout the process.
Vertical Shift
A vertical shift in a function graph involves moving the entire graph up or down without changing its shape. This is done by adding or subtracting a constant to/from the function.
In our solution, the transformation requires a shift downward by 2 units. After applying the vertical stretch, we obtained the function \(g(x) = 2x^2\). For the downward shift, we subtract 2 from this function. The resulting function is \(h(x) = 2x^2 - 2\).
Some key aspects of vertical shifts are:
  • Addition of a constant moves the graph upwards - think of it as increasing every y-value by the constant amount.
  • Subtraction of a constant moves the graph downwards, decreasing each y-value by that constant.
  • The x-values of the graph do not change during this transformation.
Familiarity with vertical shifts is crucial for graph transformations, especially when finalizing the position of a graph after other transformations.
Horizontal Shift
Horizontal shift changes the position of the graph left or right on the coordinate plane. Unlike vertical shifts, you achieve a horizontal shift by adjusting the variable inside the function. This means replacing \(x\) with \(x - c\) or \(x + c\).
Our exercise shows a horizontal shift to the right by 3 units. This is done by replacing \(x\) with \(x - 3\) in \(h(x) = 2x^2 - 2\), leading to the final function \(f(x) = 2(x - 3)^2 - 2\).
Important points about horizontal shifts include:
  • Replacing \(x\) with \(x - c\) shifts the graph to the right by \(c\) units; this may seem opposite to intuition as it involves subtracting.
  • Conversely, \(x + c\) moves the graph to the left.
  • The actual values of the function do not change; only their positions along the x-axis are affected.
Understanding horizontal shifts is crucial to navigating function transformations, ensuring precise placement of graphs on the coordinate plane.

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

In a certain country, income tax \(T\) is assessed according to the following function of income \(x\) : $$T(x)=\left\\{\begin{array}{ll} 0 & \text { if } 0 \leq x \leq 10,000 \\ 0.08 x & \text { if } 10,000

Find the domain of the function. $$f(x)=\frac{(x+1)^{2}}{\sqrt{2 x-1}}$$

Draw a machine diagram for the function. $$f(x)=\sqrt{x-1}$$

Temperature readings \(T\) (in 'F) were recorded every 2 hours from midnight to noon in Atlanta, Georgia, on March 18, 1996. The time \(t\) was measured in hours from midnight. Sketch a rough graph of \(T\) as a function of \(t\) \begin{array}{|c|c|c|c|c|c|c|c|} \hline t & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline T & 58 & 57 & 53 & 50 & 51 & 57 & 61 \\ \hline \end{array}

A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$f(x)=x^{4}-4 x^{3}+2 x^{2}+4 x-3$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.