Problem 491 - Precalculus

Question

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes. The reciprocal squared function shifted to the right 2 units.

Step-by-Step Solution

Answer

Shift \( f(x) = \frac{1}{x^2} \) right by 2 units; vertical asymptote at \( x = 2 \), horizontal asymptote at \( y = 0 \).

Step 1: Understand the Base Function

The base function for the reciprocal squared function is \( f(x) = \frac{1}{x^2} \). This function has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \).

Step 2: Apply the Horizontal Shift

To shift the function to the right by 2 units, we replace \( x \) with \( x - 2 \). The transformed function becomes \( g(x) = \frac{1}{(x - 2)^2} \).

Step 3: Identify the Vertical Asymptote

The vertical asymptote for the base function occurs where the denominator is zero. For \( g(x) = \frac{1}{(x-2)^2} \), setting \( x-2 = 0 \) gives the vertical asymptote at \( x = 2 \).

Step 4: Identify the Horizontal Asymptote

The horizontal asymptote is influenced by the behavior of the function as \( x \) approaches infinity or negative infinity. Since the transformation only involves a horizontal shift, the horizontal asymptote remains at \( y = 0 \).

Step 5: Graph the Function

Plot the function \( g(x) = \frac{1}{(x-2)^2} \). Start by marking the vertical asymptote at \( x = 2 \) and the horizontal asymptote at \( y = 0 \). Then sketch the graph, noting that it will approach the vertical asymptote as \( x \to 2 \) from either side and the horizontal asymptote as \( x \to \pm\infty \).

Key Concepts

Reciprocal Functions

Reciprocal functions are intriguing yet fundamental in mathematics. A reciprocal function takes the form of \[ f(x) = \frac{1}{x} \]These functions represent the inverse relationship between the input (\(x\)) and the output (\(y\)). For instance, as \(x\) gets larger, \(y\) becomes smaller and vice versa.
This creates a distinctive curve that is symmetric with respect to the origin, indicating it has rotational symmetry.
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• The domain of reciprocal functions excludes zero since division by zero is undefined.
• These functions often exhibit asymptotic behavior, where the graph approaches the axes but never actually touches them.

In this broader category, we encounter specific variations like the reciprocal squared function:\[ f(x) = \frac{1}{x^2} \]This alteration leads to subtle differences in graph behavior and symmetry compared to the basic reciprocal function.

Asymptotes

Asymptotes are essential components when studying the behavior of reciprocal functions. They are lines that the graph of a function approaches but never quite reaches.
Let's explore the types of asymptotes we often consider:

• Vertical Asymptotes: These occur when the function becomes undefined. For example, in the reciprocal function \( f(x) = \frac{1}{x} \), the vertical asymptote is at \( x = 0 \).
• Horizontal Asymptotes: These describe the behavior of the function as \( x \) approaches infinity or negative infinity. They tell us where the function's values settle down.

In the function \[ g(x) = \frac{1}{(x-2)^2} \]we have a vertical asymptote at \( x = 2 \) because that's where the denominator is zero. The horizontal asymptote remains at \( y = 0 \), showing the function's behavior at extreme values of \( x \).Understanding these invisible lines guides us in predicting how the function behaves across its domain.

Horizontal Shift

A horizontal shift is a type of transformation that moves the graph of a function left or right. This is done by modifying the input variable \( x \).
For example, if we have a function \( f(x) \), replacing \( x \) with \( x - h \) shifts the graph to the right by \( h \) units and replacing \( x \) with \( x + h \) shifts it to the left by \( h \) units.In our exercise, the reciprocal squared function is shifted to the right by 2 units. This transformation results in the function:\[ g(x) = \frac{1}{(x - 2)^2} \]

• The horizontal shift changes the position of vertical asymptotes. In this instance, it moves from \( x = 0 \) to \( x = 2 \).
• Horizontal asymptotes, like \( y = 0 \), are unaffected by horizontal shifts.

Graphing Functions

Graphing functions involves creating visual representations of equations on a coordinate plane. This process includes plotting the function's key points and understanding crucial features like asymptotes and intercepts.Let's investigate how you would graph the transformed function \( g(x) = \frac{1}{(x - 2)^2} \):

• Start by identifying asymptotes. In this function, the vertical asymptote is at \( x = 2 \) and the horizontal asymptote is at \( y = 0 \).
• Mark these asymptotes on your graph with dashed lines, signaling that the curve will get close to these lines but never cross them.
• Next, plot some key points. This function is negative both before and after the asymptote, so pick values of \( x \) around the asymptote like \( x = 1 \), \( x = 2.5 \), and \( x = 3 \).

Connecting these points with a smooth curve will reveal that the graph approaches the asymptotes, displaying characteristic behavior of the reciprocal squared function.