Problem 36 - 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 function shifted down one unit and left three units.

Step-by-Step Solution

Answer

Graph the function \(f(x) = \frac{1}{x+3} - 1\) with vertical asymptote at \(x = -3\) and horizontal asymptote at \(y = -1\).

Step 1: Identify the base function

The base function is the reciprocal function, which is given by \(f(x) = \frac{1}{x}\). This function has a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\).

Step 2: Apply the horizontal shift

The function is shifted left by 3 units. To account for this, we replace \(x\) with \(x+3\) in the function. Thus, the new function becomes \(f(x) = \frac{1}{x+3}\). This shift moves the vertical asymptote from \(x = 0\) to \(x = -3\).

Step 3: Apply the vertical shift

The function is shifted down by 1 unit. To account for this, we subtract 1 from the whole function, resulting in the function \(f(x) = \frac{1}{x+3} - 1\). This changes the horizontal asymptote from \(y = 0\) to \(y = -1\).

Step 4: Identify the transformed function and asymptotes

The transformed function is \(f(x) = \frac{1}{x+3} - 1\). The vertical asymptote is now \(x = -3\) and the horizontal asymptote is \(y = -1\).

Step 5: Graph the function

Graph the transformed function \(f(x) = \frac{1}{x+3} - 1\). Plot the vertical asymptote as a dashed line at \(x = -3\) and the horizontal asymptote as a dashed line at \(y = -1\). The graph of the function will approach these asymptotes but never touch them.

Key Concepts

Reciprocal Function

The reciprocal function is a fundamental concept in algebra that is often represented as \( f(x) = \frac{1}{x} \). This function is unique because it is not defined at \( x = 0 \), leading to specific behaviors and properties such as asymptotes.

One of the most important characteristics of the reciprocal function is its symmetry around both the x-axis and y-axis. This symmetry helps in understanding the graph’s behavior in different quadrants. The function decreases approaching zero as \( x \to 0^+ \), and it decreases without bound as \( x \to 0^- \). It also decreases closer to zero as \( x \to \pm \infty \).

Now, when transformations are applied, these properties shift but still maintain similar characteristics. The reciprocal function is a perfect example for learning how shifts and transformations work in general.

Vertical Asymptote

A vertical asymptote of a function occurs at a given point when a function approaches infinity or negative infinity as the x-values get closer to that specific point. For the reciprocal function \( f(x) = \frac{1}{x} \), the vertical asymptote is at \( x = 0 \). This is where the function is undefined, meaning it cannot be directly computed due to division by zero.

In the transformed function \( f(x) = \frac{1}{x+3} \), the vertical asymptote shifts according to the transformation applied. The horizontal shift left by three units moves this asymptote from \( x = 0 \) to \( x = -3 \). This reflects a general rule that, in the expression \( \frac{1}{x-c} \), the vertical asymptote is located at \( x = c \).

It’s important to plot the vertical asymptote as a dashed line to indicate where the graph of the function doesn’t cross.

Horizontal Asymptote

Horizontal asymptotes occur when the outputs of a function approach a particular value as the input grows large in either direction. The basic reciprocal function \( f(x) = \frac{1}{x} \) has a horizontal asymptote at \( y = 0 \). As \( x \) moves toward positive or negative infinity, the function values get closer and closer to zero without actually reaching it.

When we apply a vertical shift, as in \( f(x) = \frac{1}{x+3} - 1 \), we are effectively translating the graph vertically. In this case, the horizontal asymptote moves down one unit to \( y = -1 \). This vertical shift does not affect the horizontal nature of approaching a specific value but rather changes what that value will be.

Horizontal asymptotes are a key part of graphing rational functions, indicating the long-term behavior as the x-values become significantly large or small.

Function Shifts

Function shifts are transformations that change the position of a graph on the coordinate plane without altering its shape. There are two main types of shifts: horizontal and vertical.

A horizontal shift involves moving the graph left or right. In the function \( f(x) = \frac{1}{x+3} \), the graph of the basic reciprocal function \( \frac{1}{x} \) is shifted left by 3 units. This is accomplished by adding 3 to the \( x \) variable, shown as \( x+3 \).

A vertical shift moves the graph up or down. For our transformed function \( f(x) = \frac{1}{x+3} - 1 \), the graph is shifted down by 1 unit, indicated by subtracting 1 from the entire function.

These shifts are critical in modifying and understanding function graphs, as they impact the positions of asymptotes and overall graph appearance while maintaining the intrinsic shape and behavior of the original function.