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Chapter 7: Problem 9

Begin with the function \(f(x)=x^{-3}\). Then: a. Create a new function \(g(x)\) by vertically stretching \(f(x)\) by a factor of 4 b. Create a new function \(h(x)\) by vertically compressing \(f(x)\) by a factor of \(\frac{1}{2}\). c. Create a new function \(j(x)\) by first vertically stretching \(f(x)\) by a factor of 3 and then by reflecting the result across the \(x\) -axis.

Short Answer

Expert verified
a. \(g(x) = 4x^{-3}\); b. \(h(x) = \frac{1}{2}x^{-3}\); c. \(j(x) = -3x^{-3}\)

Step by step solution

01

Given Function

Consider the given function: \[ f(x) = x^{-3} \]
02

Vertical Stretch by a Factor of 4

To create the new function \(g(x)\) by vertically stretching \(f(x)\) by a factor of 4, multiply the entire function by 4: \[ g(x) = 4 \times f(x) = 4 \times x^{-3} = 4x^{-3} \]
03

Vertical Compression by a Factor of 1/2

To create the new function \(h(x)\) by vertically compressing \(f(x)\) by a factor of \(\frac{1}{2}\), multiply the entire function by \(\frac{1}{2}\): \[ h(x) = \frac{1}{2} \times f(x) = \frac{1}{2} \times x^{-3} = \frac{1}{2}x^{-3} \]
04

Vertical Stretch by a Factor of 3 and Reflection Across the x-axis

To create the new function \(j(x)\) by first vertically stretching \(f(x)\) by a factor of 3 and then reflecting it across the x-axis, first multiply the function by 3 and then multiply by -1 to reflect: \[ j(x) = - \left(3 \times f(x)\right) = - \left(3 \times x^{-3}\right) = -3x^{-3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Stretch
When we **vertically stretch** a function, we multiply the function by a constant factor greater than 1. This makes the graph of the function taller and narrower, with each point on the graph moving farther from the x-axis.

For example, given the function \( f(x) = x^{-3} \), and we want to stretch it vertically by a factor of 4, we multiply the entire function by 4. Thus, the new function will be \( g(x) = 4f(x) \).

So, \[ g(x) = 4x^{-3} \]

This transformation affects the y-values of the function. Each y-value is now four times larger than in the original function, making the curve appear taller.

To identify a vertical stretch:
  • Ensure the stretching factor is greater than 1.
  • Multiply the entire original function by this factor.
Vertical Compression
A **vertical compression** involves reducing the height of the graph by multiplying the function by a constant factor between 0 and 1. This makes the graph shorter and wider, bringing the points closer to the x-axis.

For instance, if we take the function \( f(x) = x^{-3} \) and vertically compress it by a factor of \(\frac{1}{2}\), we need to multiply the entire function by \(\frac{1}{2}\). The revised function will be:

\[ h(x) = \frac{1}{2} f(x) = \frac{1}{2} x^{-3} \]

This operation impacts the y-values, making each y-value half of what it was originally, flattening the curve.

To recognize a vertical compression:
  • The compression factor will be between 0 and 1.
  • Multiply the entire original function by this factor.
Reflection Across the x-axis
Reflecting a function **across the x-axis** means that every point of the function is flipped over the x-axis. This transformation changes the sign of the function's y-values, turning positive y-values into negative ones and vice versa.

Let's consider the base function \( f(x) = x^{-3} \). If we want to stretch it vertically by a factor of 3 and then reflect it across the x-axis, we first multiply by 3 to create the vertical stretch:

\[ j(x) = 3 f(x) = 3 x^{-3} \]

Next, to reflect it across the x-axis, multiply the function by -1:

\[ j(x) = - (3 x^{-3}) = -3 x^{-3} \]

This transformation affects all the y-values by making them their negative counterparts, essentially flipping the graph upside down.

To perform a reflection across the x-axis:
  • Multiply the entire original function by -1.

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Most popular questions from this chapter

Assume \(Y\) is directly proportional to \(X^{3}\). a. Express this relationship as a function where \(Y\) is the dependent variable. b. If \(Y=10\) when \(X=2\), then find the value of the constant of proportionality in part (a). c. If \(X\) is increased by a factor of 5 , what happens to the value of \(Y ?\) d. If \(X\) is divided by \(2,\) what happens to the value of \(Y ?\) e. Rewrite your equation from part (a), solving for \(X .\) Is \(X\) directly proportional to \(Y ?\)

The frequency, \(F\) (the number of oscillations per unit of time), of an object of mass \(m\) attached to a spring is inversely proportional to the square root of \(m\). a. Write an equation describing the relationship. b. If a mass of \(0.25 \mathrm{~kg}\) attached to a spring makes three oscillations per second, find the constant of proportionality, c. Find the number of oscillations per second made by a mass of \(0.01 \mathrm{~kg}\) that is attached to the spring discussed in part (b).

The radius of Earth is about 6400 kilometers. Assume that Earth is spherical. Express your answers to the questions below in scientific notation. a. Find the surface area of Earth in square meters. b. Find the volume of Earth in cubic meters. c. Find the ratio of the surface area to the volume.

(Graphing program optional.) Plot the functions \(f(x)=x^{2}\), \(g(x)=3 x^{2},\) and \(h(x)=\frac{1}{4} x^{2}\) on the same grid. Insert the symbol \(>\) or \(<\) to make the relation true. a. For \(x>0, g(x) \quad f(x) __________ h(x)\) b. For \(x<0, g(x) \quad f(x) __________ h(x)\)

A graphing program is useful for many of the exercises in this section. Sketch by hand and compare the graphs of the following: $$ \begin{array}{lll} y_{1}=x^{2} & y_{2}=-x^{2} & y_{3}=2 x^{2} \\ y_{4}=-2 x^{2} & y_{5}=\frac{1}{2} x^{2} & y_{6}=-\frac{1}{2} x^{2} \end{array} $$
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