Problem 38 - 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 down 2 units and right 1 unit.

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 1 \); horizontal asymptote at \( y = -2 \).

Step 1: Understand the Base Function

The base function for this transformation is the reciprocal squared function, which is given by \( f(x) = \frac{1}{x^2} \). Its graph includes vertical asymptotes at \( x = 0 \) and a horizontal asymptote at \( y = 0 \).

Step 2: Apply the Horizontal Shift

The function is shifted right by 1 unit. To account for this transformation, replace \( x \) with \( x - 1 \). This results in the new function \( f(x) = \frac{1}{(x-1)^2} \). The vertical asymptote moves from \( x = 0 \) to \( x = 1 \).

Step 3: Apply the Vertical Shift

The function is shifted down by 2 units. Subtract 2 from the previous step's result. The updated function is \( f(x) = \frac{1}{(x-1)^2} - 2 \). This affects the horizontal asymptote, moving it from \( y = 0 \) to \( y = -2 \).

Step 4: Note the Asymptotes

After applying both transformations, the function \( f(x) = \frac{1}{(x-1)^2} - 2 \) has a vertical asymptote at \( x = 1 \) and a horizontal asymptote at \( y = -2 \).

Key Concepts

Reciprocal Squared Function

A reciprocal squared function is a unique mathematical function that takes the form \( f(x) = \frac{1}{x^2} \). This function is interesting because its values depend on the reciprocal of the square of \( x \). As \( x \) approaches zero, the denominator grows smaller, causing the function's value to increase sharply.

• The graph of this function is always positive, since squaring any real number yields a non-negative result.
• It features a U-shaped curve, symmetric about the y-axis due to having even powers of \( x \).

When transforming such a function, key characteristics like its asymptotes and general shape provide crucial guidance. Understand how each transformation properly alters the graph to predict the final look accurately.

Vertical Asymptote

Vertical asymptotes are lines where a function approaches infinity; they signal non-existence for real values at specific \( x \)-values. In the reciprocal squared function \( f(x) = \frac{1}{x^2} \), the vertical asymptote naturally occurs at \( x = 0 \), because dividing by zero is undefined.

• When applying horizontal shifts, like a move right by 1 unit, the vertical asymptote shifts as well. So our modified function \( f(x) = \frac{1}{(x-1)^2} \) has a vertical asymptote at \( x = 1 \).
• This transformation doesn't alter the infinite qualities of the curve near the asymptote but merely dictates the specific \( x \)-value not included in the domain.

Locating vertical asymptotes is essential in sketching rational function graphs since they reveal critical points of interest and behavior as \( x \) approaches those values.

Horizontal Asymptote

Horizontal asymptotes indicate the value a function tends to as \( x \) becomes very large or very small. In reciprocal squared functions like \( f(x) = \frac{1}{x^2} \), the horizontal asymptote is \( y = 0 \) because as \( x \) increases, \( \frac{1}{x^2} \) diminishes to zero.

• After a vertical shift occurs, the entire function graph moves parallel to the y-axis. For the example shift down by 2 units in \( f(x) = \frac{1}{(x-1)^2} - 2 \), makes the horizontal asymptote appear at \( y = -2 \).
• This shift doesn't change the shape of the curve, yet it lowers the asymtote's nobility as \( x \) elongates in magnitude.

Understanding the role of horizontal asymptotes in predicting long-term behavior clarifies how functions extend into the indefinite right and left directions on the Cartesian plane.