Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{1}{x+2}$$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{1}{x+2}\) has a vertical asymptote at \(x=-2\) and a horizontal asymptote at \(y=0\). It does not have any x-intercepts, but it does have a y-intercept at \(y=\frac{1}{2}\). The function is undefined at \(x=-2\).

Step 1: Identify the Vertical Asymptote

Start by identifying the vertical asymptote. This is the value of \(x\) that makes the denominator of the function equal to zero. In this case, it's \(x+2=0\), so \(x=-2\) is the vertical asymptote.

Step 2: Identify the horizontal asymptote

Next, determine the horizontal asymptote. For a function of the form \(f(x)=\frac{1}{x+n}\), the horizontal asymptote is always \(y=0\). So, in the given function, the horizontal asymptote is \(y=0\). It means that as \(x\) approaches positive or negative infinity, \(f(x)\) will approach zero.

Step 3: Identify the Intercepts

The x-intercept is the value of \(x\) that makes \(f(x)\) equal to zero. But since there's no real \(x\) which makes \(f(x)=\frac{1}{x+2}\) equal to zero, there's no x-intercept. The y-intercept is the value of \(f(x)\) when \(x=0\). Substituting \(x=0\) into the equation gives \(f(0)=\frac{1}{0+2}=\frac{1}{2}\), showing that \(y=\frac{1}{2}\) is the y-intercept.

Step 4: Sketch the Graph

Finally, sketch the graph. As \(x\) approaches the vertical asymptote (\(x=-2\)) from left or right, \(f(x)\) approaches positive or negative infinity respectively. The curve should get closer and closer to the horizontal asymptote (line \(y=0\)) as \(x\) moves away from \(x=-2\) but should never touch or cross it. Indicate the y-intercept (\(y=\frac{1}{2}\)) on the graph.

Key Concepts

Understanding Vertical Asymptotes

When sketching rational functions, one of the key features to look for are the vertical asymptotes. These are the lines to which the function will get infinitely close but never actually touch or cross. In simpler terms, they are like invisible boundaries that the graph cannot cross.

The vertical asymptote occurs at values of x that make the denominator of the rational function equal to zero. In our example, the function is given as \(f(x)=\frac{1}{x+2}\). To find the vertical asymptote, you'll set the denominator equal to zero and solve for x. This gives us \(x+2=0\) or \(x=-2\), which is the equation of the vertical asymptote in this case.

It's crucial to remember that as x approaches the value of the vertical asymptote, the function's value will tend to become infinitely large or small (depending on the direction from which x is approaching). The graph will show a sharp descent or ascent as it nears the line x=-2. However, it's important to note that while the function does not have a value at \(x=-2\), it can be approached as closely as desired on either side.

Grasping Horizontal Asymptotes

Another aspect of rational functions to understand is their horizontal asymptotes. Unlike vertical asymptotes, which indicate where the function is undefined, horizontal asymptotes help us understand the behavior of a function as x approaches infinity or negative infinity. In more human terms, they show us where the function is headed in the long run as we move far to the left or right along the x-axis.

For the function \(f(x) = \frac{1}{x+2}\), the horizontal asymptote is found by observing the degrees of the polynomials in the numerator and the denominator. Here, the degree of the numerator (0, since there's just a constant 1) is less than the degree of the denominator (1, for x). When this happens, the horizontal asymptote is always at \(y=0\). This tells us that no matter how far we move along the x-axis, the value of the function will always approach zero; the graph will never touch or cross this horizontal line but will get infinitely closer to it.

Finding the Y-intercept

The y-intercept is a point where the graph of the function crosses the y-axis. It's the output value of the function when the input value x is zero. To find it, we substitute \(x=0\) into the rational function and solve for f(x).

With our example, \(f(x)=\frac{1}{x+2}\), we find the y-intercept by calculating \(f(0)=\frac{1}{0+2}=\frac{1}{2}\). This means that the point \(0, \frac{1}{2}\) is where the graph intersects the y-axis. This intercept provides a specific point which is useful for plotting the graph accurately. It's especially helpful since it tells us where to start or 'enter the scene' on the graph when sketching out our rational function's curve.