Problem 492 - 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 down 2 units and right 1 unit.

Step-by-Step Solution

Answer

The vertical asymptote is \( x = 1 \) and the horizontal asymptote is \( y = -2 \).

Step 1: Identify the Parent Function

The parent function here is the reciprocal squared function, which is given by \( f(x) = \frac{1}{x^2} \). This is the function we start with before applying any transformations.

Step 2: Apply Horizontal Shift

The function is shifted to the right by 1 unit. For horizontal shifts, we modify the argument inside the function: \( f(x) = \frac{1}{(x-1)^2} \). This moves the graph one unit to the right.

Step 3: Apply Vertical Shift

The function is also shifted down by 2 units. This affects the whole function: \( f(x) = \frac{1}{(x-1)^2} - 2 \). This moves the graph 2 units downward.

Step 4: Identify Vertical Asymptote

A vertical asymptote occurs where the denominator is zero since the function is undefined at this point. In \( f(x) = \frac{1}{(x-1)^2} - 2 \), set \( (x-1)^2 = 0 \), giving the vertical asymptote at \( x = 1 \).

Step 5: Identify Horizontal Asymptote

For horizontal asymptotes, observe the end behavior of the function. As \( x \) approaches infinity, \( \frac{1}{(x-1)^2} \to 0 \), hence the horizontal asymptote is where the function approaches: \( y = -2 \).

Step 6: Graph the Function

Plot the graph based on the identified shifts and asymptotes. It has a vertical asymptote at \( x = 1 \), a horizontal asymptote at \( y = -2 \), and shifts accordingly: down 2 units and right 1 unit from the basic function.

Key Concepts

Reciprocal Function

A reciprocal function refers to a function that contains a ratio of 1 divided by a function, such as the reciprocal squared function in this exercise, defined as \( f(x) = \frac{1}{x^2} \). This function has a characteristic curve where the graph approaches but never touches the x-axis or the vertical axis, exhibiting vertical and horizontal asymptotes. The behavior of this function largely depends on the transformations applied to it. The reciprocal squared function is particularly noted for its steep decline as \( x \) moves away from zero in either direction.

Horizontal Shift

A horizontal shift involves moving the graph of a function left or right on the coordinate plane. In our case with the reciprocal squared function \( f(x) = \frac{1}{x^2} \), a transformation to \( f(x) = \frac{1}{(x-1)^2} \) results in a horizontal shift to the right by 1 unit.
This shift changes the position of the graph without altering its shape or orientation, meaning the key features such as asymptotes will also shift correspondingly.

• If \( x \) is replaced with \( x - c \), the graph shifts \( c \) units to the right.
• If \( x \) is replaced with \( x + c \), the graph shifts \( c \) units to the left.

The horizontal shift plays a crucial role in determining where the vertical asymptote of a function will be located, in this case at \( x = 1 \).

Vertical Shift

A vertical shift moves a function's graph up or down. For the reciprocal squared function, after the horizontal shift has been applied, the transformation \( f(x) = \frac{1}{(x-1)^2} - 2 \) denotes a vertical shift downward by 2 units.

• When a constant \( c \) is subtracted from the function, the graph shifts downward by \( c \) units.
• If a constant is added, the graph moves upward by that constant.

The vertical shift directly affects the horizontal asymptote instead of altering the vertical one. Thus when shifted down by 2 units, the horizontal asymptote, which initially was at \( y = 0 \), now moves to \( y = -2 \).

Asymptotes

Asymptotes represent lines that a graph approaches but never actually reaches. In transformation of functions, identifying asymptotes is critical to understanding the behavior of the graph at its extremities and certain points.

• Vertical Asymptotes: Occur where the function is undefined. For \( f(x) = \frac{1}{(x-1)^2} - 2 \), it occurs at \( x = 1 \) due to division by zero in the denominator.
• Horizontal Asymptotes: Describe the end behavior of the function. As x approaches infinity, the terms impacting y diminish. Thus, the function approaches \( y = -2 \).

Asymptotes are important landmarks when graphing functions as they help delimit the graph's range, guide the shape, and predict long-term behavior.