Problem 37 - College Algebra with Corequisite Support

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

Graph \( g(x) = \frac{1}{(x-2)^2} \) with vertical asymptote at \( x=2 \) and horizontal asymptote at \( y=0 \).

Step 1: Identify the Base Function

The reciprocal squared function is given by \( f(x) = \frac{1}{x^2} \). This function has vertical asymptotes at \( x=0 \) and approaches zero as \( x \to \pm\infty \), giving it a horizontal asymptote at \( y=0 \).

Step 2: Apply Horizontal Transformation

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

Step 3: Determine the New Vertical Asymptote

The vertical asymptote of the original function \( f(x) = \frac{1}{x^2} \) is at \( x=0 \). After shifting 2 units to the right, the vertical asymptote moves to \( x=2 \).

Step 4: Confirm the Horizontal Asymptote

The horizontal asymptote for both the base and transformed functions is \( y=0 \), as \( g(x) = \frac{1}{(x-2)^2} \to 0 \) as \( x \to \pm\infty \).

Step 5: Graph the Transformed Function

Graph \( g(x) = \frac{1}{(x-2)^2} \) by plotting several points to verify its shape, placing the vertical asymptote at \( x=2 \) and the horizontal asymptote at \( y=0 \). Highlight these asymptotes on the graph.

Key Concepts

Reciprocal Function

The reciprocal function is a fascinating mathematical function often used in transformations. In its basic form, the reciprocal function is defined as \( f(x) = \frac{1}{x} \). This function is known for two key characteristics:

• It features a vertical asymptote at \( x=0 \). This occurs because the function is undefined where the denominator is zero.
• It also features a horizontal asymptote at \( y=0 \), as the function value approaches zero as \( x \) grows larger in magnitude (either positively or negatively).

When we talk about a reciprocal squared function, like \( f(x) = \frac{1}{x^2} \), it inherits similar properties but with slightly altered characteristics. For this function, the vertical asymptote remains at \( x=0 \). However, because all outputs are non-negative, the graph exists only above the \( x \)-axis. In transformations, we might shift this whole graph horizontally or vertically, but the principles of asymptotic behavior remain core to understanding its graph.

Horizontal Asymptote

A horizontal asymptote is a constant value that a function approaches as the variable \( x \) moves toward positive or negative infinity. In the context of the reciprocal squared function \( f(x) = \frac{1}{x^2} \), the horizontal asymptote is \( y=0 \).
This happens because as \( x \) becomes very large or very negative, the value of \( f(x) \) becomes exceedingly small, approaching zero but never actually reaching it. The shift \( (x-2) \) does not affect the horizontal asymptote. Whether or not the graph shifts horizontally, the approach towards zero remains unaffected.
Understanding horizontal asymptotes in reciprocal functions helps us predict the end behavior of the graph. No matter how the graph shifts, its tails will aim closer and closer to the \( x \)-axis. This feature is invaluable for sketching and interpreting the behavior of reciprocal squared functions.

Vertical Asymptote

Vertical asymptotes are lines that the graph of a function will approach but never touch or intersect. For the reciprocal squared function \( f(x) = \frac{1}{x^2} \), we initially have a vertical asymptote at \( x=0 \). This is because the function becomes undefined at this point.
When transformations are applied, such as a horizontal shift to the right by 2 units, denoted by \( g(x) = \frac{1}{(x-2)^2} \), the vertical asymptote also shifts. Instead of being at \( x=0 \), it moves to \( x=2 \). This recognizes that the point of discontinuity in the function has shifted, staying consistent with the operation inside the function's argument.
Vertical asymptotes help pinpoint the behavior of functions near discontinuities, demonstrating how they shoot up towards infinity or drop down towards negative infinity. They are crucial in understanding the architecture of graph transformations, particularly for reciprocal functions.