Problem 50 - Thomas Calculus

Question

Graph the functions in Exercises \(35-54\) $$ y=\frac{1}{x+2} $$

Step-by-Step Solution

Answer

The function has a vertical asymptote at \( x = -2 \) and a horizontal asymptote at \( y = 0 \). Graph resembles a hyperbola.

Step 1: Identify Asymptotes

The function given is \( y = \frac{1}{x+2} \), which is a rational function. First, identify vertical asymptotes by setting the denominator equal to zero, \( x + 2 = 0 \), yielding \( x = -2 \). There is a vertical asymptote at \( x = -2 \). Since this function is a basic rational function of the form \( \frac{1}{x+c} \), the horizontal asymptote is \( y = 0 \).

Step 2: Determine Key Points

To plot the graph more accurately, calculate a few key points by substituting values of \( x \) into the function. For example:- \( x = -3 \rightarrow y = \frac{1}{-3+2} = -1 \)- \( x = -1 \rightarrow y = \frac{1}{-1+2} = 1 \)- \( x = 0 \rightarrow y = \frac{1}{0+2} = 0.5 \)- \( x = -4 \rightarrow y = \frac{1}{-4+2} = -0.5 \)

Step 3: Plot the Graph

On a graph, plot the vertical asymptote as a dashed line at \( x = -2 \) and the horizontal asymptote as a dashed line at \( y = 0 \). Then, plot the points calculated in Step 2. The graph approaches the asymptotes but never touches them. It should resemble a hyperbola with the vertical asymptote dividing the graph into two separate parts.

Step 4: Analyze Graph Behavior

As \( x \) approaches \( -2 \) from the left, \( y \) approaches negative infinity, and as \( x \) approaches \( -2 \) from the right, \( y \) approaches positive infinity. Both arms of the graph approach the horizontal asymptote \( y = 0 \) as \( x \) goes to plus or minus infinity.

Key Concepts

Asymptotes

Understanding asymptotes is crucial when dealing with rational functions. In the case of the function \( y = \frac{1}{x+2} \), asymptotes provide essential information about where the graph "heads off to infinity" or levels out.
Vertical asymptotes occur when the denominator of a rational function equals zero, resulting in an undefined value. For our function, set the denominator \( x+2 = 0 \), which results in \( x = -2 \). This means the graph has a vertical asymptote at \( x = -2 \). Vertical asymptotes tell us that the graph will extend infinitely far up or down as it gets close to this line but will never actually touch it.
Horizontal asymptotes, on the other hand, indicate a line that the graph will approach as it goes towards infinity in either direction along the x-axis. In our specific example, the function is of the form \( \frac{1}{x+c} \), which typically has a horizontal asymptote at \( y = 0 \). This gives us an understanding of the graph's behavior, especially at far-left and far-right ends of the x-axis.

Graphing Functions

Graphing rational functions can be a straightforward yet informative process. With the function \( y = \frac{1}{x+2} \), we use the asymptotes as guides.
Start by drawing dashed lines where the vertical and horizontal asymptotes occur, at \( x = -2 \) and \( y = 0 \) respectively. These lines help set the boundaries of where the function will behave in an extreme manner.

• As the input \( x \) gets very close to \( x = -2 \) from either side, observe how \( y \) changes dramatically towards positive or negative infinity.
• As \( x \) becomes very large or very small (positive or negative), the output \( y \) will get closer to zero but never settle quite on it.

After plotting these asymptotes, take your key points, which provide exact locations to place on the graph. These points, found by plugging chosen \( x \) values into the function, act as checkpoints. Remember, the overall shape is similar to a hyperbola, curving towards the asymptotes but not touching them.

Key Points

In determining the precise shape of a rational function's graph, key points are indispensable. These points are specific solutions to the function that you calculate by inputting various \( x \) values.
For instance, using the function \( y = \frac{1}{x+2} \), we can substitute particular values like \( x = -3, -1, 0, \) and \( -4 \) to get corresponding \( y \) values:

• At \( x = -3 \), \( y = -1 \).
• At \( x = -1 \), \( y = 1 \).
• At \( x = 0 \), \( y = 0.5 \).
• At \( x = -4 \), \( y = -0.5 \).

Plot these points on your graph to capture a more realistic curving effect approaching asymptotes. They illustrate the behavior of the curve in different regions, showing accurately how it bends and routes itself around the asymptotes without touching them. This precision ensures that our visual representation faithfully reflects the function's characteristics.