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Chapter 7: Problem 20

(Graphing program required.) Begin with a function \(f(x)=\frac{1}{x^{3}}\). In each part describe how the graph of \(g(x)\) is related to the graph of \(f(x)\). Support your answer by graphing each function with technology. a. \(g(x)=\frac{2}{x^{3}}\) b. \(g(x)=-\frac{1}{x^{3}}\) c. \(g(x)=-\frac{1}{2} x^{-3}\)

Short Answer

Expert verified
a. Vertical stretch by 2. b. Reflection about x-axis. c. Reflection and vertical compression by \( \frac{1}{2} \).

Step by step solution

01

Understand the Base Function

The base function provided is \( f(x)=\frac{1}{x^{3}} \). This is a reciprocal power function, and its graph has specific characteristics such as being undefined at \( x = 0 \) and having a vertical asymptote at \( x = 0 \). For positive values of \( x \), the function decreases as \( x \) increases, and for negative values of \( x \), the function increases as \( x \) becomes more negative.
02

Analyze and Graph \( g(x)=\frac{2}{x^{3}} \)

The function \( g(x)=\frac{2}{x^{3}} \) involves multiplying the base function by 2. This results in a vertical stretch of the graph by a factor of 2. To illustrate this, graph both \( f(x)=\frac{1}{x^{3}} \) and \( g(x)=\frac{2}{x^{3}} \) on the same coordinate plane and observe that the values of \( g(x) \) are twice as large as those of \( f(x) \) for any \( x \).
03

Analyze and Graph \( g(x)=-\frac{1}{x^{3}} \)

The function \( g(x)=-\frac{1}{x^{3}} \) involves multiplying the base function by -1. This results in a reflection of the graph about the x-axis. Graph both \( f(x)=\frac{1}{x^{3}} \) and \( g(x)=-\frac{1}{x^{3}} \) on the same coordinate plane and observe that for any \( x \), the values of \( g(x) \) are the opposites of those of \( f(x) \).
04

Analyze and Graph \( g(x)=-\frac{1}{2} x^{-3} \)

First, note that \( x^{-3} = \frac{1}{x^{3}} \), so \( g(x)=-\frac{1}{2} x^{-3} = -\frac{1}{2} \cdot \frac{1}{x^{3}} = -\frac{1}{2x^{3}} \). This function involves multiplying the base function by -\( \frac{1}{2} \). Thus, it is a reflection about the x-axis (due to the negative sign) and a vertical compression by a factor of \( \frac{1}{2} \). Graph \( f(x)=\frac{1}{x^{3}} \) and \( g(x)=-\frac{1}{2x^{3}} \) on the same coordinate plane to observe these transformations.

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Key Concepts

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

Vertical Stretch
A vertical stretch occurs when a function is multiplied by a factor greater than 1. This transformation increases the distance of each point on the graph from the x-axis, effectively 'stretching' the graph vertically. For example, starting with the function \(f(x)=\frac{1}{x^{3}}\), if we modify it to \(g(x)=\frac{2}{x^{3}}\), we are multiplying by 2. This factor of 2 stretches the graph of the original function vertically. Every y-value in \(g(x)\) is twice as large as in \(f(x)\).

To visualize this transformation, graph both functions on the same set of axes. You will see that the graph of \(g(x)\) extends further away from the x-axis compared to the graph of \(f(x)\). This is a clear representation of how vertical stretching affects a function.
Reflection about the x-axis
A reflection about the x-axis involves multiplying a function by -1. This flips the graph over the x-axis, so points above the x-axis move below it and vice versa. For instance, with the reciprocal power function \(f(x)=\frac{1}{x^{3}}\), changing it to \(g(x)=-\frac{1}{x^{3}}\) results in a reflection.

When plotted, both functions will share a common shape, but \(g(x)\) will be an upside-down version of \(f(x)\). If \(f(x)\) had a point at (1,1), \(g(x)\) will have a point at (1,-1). The transformation is simple yet powerful, showing how a negative multiplier impacts the function's graph.
Vertical Compression
Vertical compression occurs when a function is multiplied by a factor between 0 and 1. This transformation 'compresses' the graph towards the x-axis, making it appear flatter. Consider \(f(x)=\frac{1}{x^{3}}\) and transform it to \(g(x)=-\frac{1}{2} x^{-3}\). This can be re-written as \(g(x)=-\frac{1}{2x^{3}}\).

Firstly, the negative sign means we reflect the graph about the x-axis. Secondly, the factor of 0.5 compresses the graph vertically. As a result, \(g(x)\) will be a reflected, compressed version of \(f(x)\). The y-values of \(g(x)\) will be half of those of \(f(x)\) in magnitude, but in the opposite direction. Graph these functions together to observe this combined transformation visually.

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Most popular questions from this chapter

Given the points (1,2) and (6,72): a. Find a linear function of the form \(y=a x+b\) that goes through the two points. b. Find an exponential function of the form \(y=a b^{x}\) that goes through the two points. c. Find a power function of the form \(y=a x^{p}\) that goes through the two points.

If the radius of a sphere is \(x\) meters, what happens to the surface area and to the volume of a sphere when you: a. Quadruple the radius? b. Multiply the radius by \(n\) ? c. Divide the radius by 3 ? d. Divide the radius by \(n\) ?

Using rules of logarithms, convert each equation to its power function equivalent in the form \(y=k x^{p}\). a. \(\log y=\log 4+2 \log x\) c. \(\log y=\log 1.25+4 \log x\) b. \(\log y=\log 2+4 \log x\) d. \(\log y=\log 0.5+3 \log x\)

Create an equation that meets the given specifications and then solve for the indicated variable. a. If \(P\) is inversely proportional to \(f\), and \(P=0.16\) when \(f=0.1,\) then what is the value for \(P\) when \(f=10 ?\) b. If \(Q\) is inversely proportional to the square of \(r\), and \(Q=6\) when \(r=3\), then what is the value for \(Q\) when \(r=9 ?\) c. If \(S\) is inversely proportional to \(w\) and \(p\), and \(S=8\) when \(w=4\) and \(p=\frac{1}{2}\), what is the value for \(S\) when \(w=8\) and \(p=1 ?\) d. \(W\) is inversely proportional to the square root of \(u\) and \(W=\frac{1}{3}\) when \(u=4\). Find \(W\) when \(u=16\)

The cost, \(C,\) in dollars, of insulating a wall is directly proportional to the area, \(A,\) of the wall (measured in \(\mathrm{ft}^{2}\) ) and the thickness, \(t,\) of the insulation (measured in inches). a. Write a cost equation for insulating a wall. b. If the insulation costs are \(\$ 12\) when the area is \(50 \mathrm{ft}^{2}\) and the thickness is 4 inches, find the constant of proportionality. c. A storage room is \(15 \mathrm{ft}\) by \(20 \mathrm{ft}\) with \(8-\mathrm{ft}\) -high ceilings. Assuming no windows and one uninsulated door that is 3 feet by 7 feet, what is the total area of the insulated walls for this room? d. What is the cost of insulating this room with 4 -inch insulation? With 6 -inch insulation?
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