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Chapter 3: Problem 58

Graph the functions by using transformations of the graphs of \(y=\frac{1}{x}\) and \(y=\frac{1}{x^{2}}\). $$ k(x)=\frac{1}{x^{2}}-3 $$

Short Answer

Expert verified
Translate \(y = \frac{1}{x^{2}}\) down by 3 units. The new function is \(k(x) = \frac{1}{x^{2}} - 3\).

Step by step solution

01

Identify the Base Function

The base function provided is \(y = \frac{1}{x^{2}}\). This function has a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\). It is already graphed based on these properties.
02

Understand the Transformation

The given function for transformation is \(k(x) = \frac{1}{x^{2}} - 3\). Notice that this is the base function \(y = \frac{1}{x^{2}}\) with a vertical translation downward by 3 units due to the \(-3\).
03

Apply the Vertical Translation

To apply the vertical translation, shift every point on the graph of \(y = \frac{1}{x^{2}}\) down by 3 units. This means that if a point was originally at \((a, b)\), it will now be at \((a, b-3)\).
04

Identify the New Asymptotes

After shifting the graph downward, the new horizontal asymptote will now be at \(y=-3\) instead of \(y=0\). The vertical asymptote remains at \(x=0\) because the translation does not affect the x-values.
05

Graph the Transformed Function

Plot the new graph using the adjusted points and asymptotes. The original graph of \(y = \frac{1}{x^{2}}\) is shifted down by 3 units, and the key characteristics of the graph should reflect this transformation.

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Key Concepts

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

Vertical Translation
Vertical translation is a type of transformation that moves the graph of a function up or down without changing its shape. In our example, we start with the base function, which is \(y = \frac{1}{x^{2}}\). When we look at \(k(x) = \frac{1}{x^{2}}-3\), we see that the graph is shifted vertically.

This means that every point on the graph is moved downward by 3 units. For instance, if a point on the original graph is at \((a, b)\), it will move to \((a, b-3)\). This shifts the graph lower, but the overall shape remains the same.

To visualize this, let's consider a few points:
  • The point \((1, 1)\) on \(y = \frac{1}{x^{2}}\) moves to \((1, -2)\).

  • The point \((2, 0.25)\) on \(y = \frac{1}{x^{2}}\) moves to \((2, -2.75)\).
By shifting the graph down, we create a new graph for \(k(x)\) that reflects this vertical translation.
Asymptotes
Asymptotes are lines that the graph of a function approaches but never actually touches. They serve as boundaries for the graph.

For the base function \(y = \frac{1}{x^{2}}\), there are two key asymptotes to note:
  • A vertical asymptote at \(x=0\).
  • A horizontal asymptote at \(y=0\).
When we translate the graph vertically, the horizontal asymptote also moves. In the case of \(k(x) = \frac{1}{x^{2}} - 3\), the horizontal asymptote is shifted down by 3 units, now sitting at \(y = -3\). The vertical asymptote remains unchanged at \(x = 0\) since the translation doesn't affect the x-values.

Understanding these new boundaries helps in accurately graphing the transformed function and understanding its behavior near these critical lines.
Base Function Transformation
Transforming a base function involves altering its graph through translations, stretches, compressions, or reflections. In this exercise, we're specifically dealing with a vertical translation.

The base function provided is \(y = \frac{1}{x^{2}}\). This function has:
  • A vertical asymptote at \(x=0\).
  • A horizontal asymptote at \(y=0\).
Our task is to transform this base function into the given function \(k(x) = \frac{1}{x^{2}} - 3\). The transformation process is straightforward:
  • Identify the base function's properties and graph it.
  • Apply the vertical translation by shifting the entire graph 3 units down.
  • Adjust the asymptotes based on the vertical shift.
By following these steps, the transformed function is graphically represented while maintaining the core behavior dictated by the base function's transformation.

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Most popular questions from this chapter

A professional fireworks team shoots an 8 -in. mortar straight upwards from ground level with an initial velocity of \(216 \mathrm{ft} / \mathrm{sec} .\) (See Example 6\()\) a. Write a function modeling the vertical position \(s(t)\) (in \(\mathrm{ft}\) ) of the shell at a time \(t\) seconds after launch. b. The mortar is designed to explode when the shell is at its maximum height. How long after launch will the shell explode? (Hint: Consider the vertex formula from Section 3.1.) c. The spectators can see the shell rising once it clears a 200 -ft tree line. For what period of time after launch is the shell visible before it explodes?

A rectangular quilt is to be made so that the length is 1.2 times the width. The quilt must be between \(72 \mathrm{ft}^{2}\) and \(96 \mathrm{ft}^{2}\) to cover the bed. Determine the restrictions on the width so that the dimensions of the quilt will meet the required area. Give exact values and the approximated values to the nearest tenth of a foot.

Write an equation of a function that meets the given conditions. Answers may vary. \(x\) -intercepts: (-3,0) and (-1,0) vertical asymptote: \(x=2\) horizontal asymptote: \(y=1\) \(y\) -intercept: \(\left(0, \frac{3}{4}\right)\)

Write the domain of the function in interval notation. $$ p(x)=\sqrt{2 x^{2}+9 x-18} $$

Sometimes it is necessary to use a "friendly" viewing window on a graphing calculator to see the key features of a graph. For example, for a calculator screen that is 96 pixels wide and 64 pixels high, the "decimal viewing window" defined by [-4.7,4.7,1] by [-3.1,3.1,1] creates a scaling where each pixel represents 0.1 unit. The window [-9.4,9.4,1] by [-6.2,6.2,1] defines each pixel as 0.2 unit, and so on. Exercises \(112-113\) compare the use of the standard viewing window to a "friendly" viewing window. a. Identify any vertical asymptotes of the function defined by \(f(x)=\frac{x^{2}+3 x+2}{x+1}\). b. Compare the graph of \(f(x)=\frac{x^{2}+3 x+2}{x+1}\) on the standard viewing window [-10,10,1] by [-10,10,1 and on the window [-4.7,4.7,1] by [-3.1,3.1,1] . Which graph shows the behavior at \(x=-1\) more completely?
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