Problem 12 - The Calculus with Analytic Geometry

Question

Find the horizontal and vertical asymptotes of the graph of the function defined by the given equation, and draw a sketch of the graph. $$ h(x)=\frac{x}{\sqrt{x^{2}-9}} $$

Step-by-Step Solution

Answer

Vertical asymptotes: \( x = ±3 \). Horizontal asymptote: \( y = 1 \).

Step 1 - Identify the vertical asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is nonzero. Set the denominator equal to zero and solve for the values of x: \ \ \( \sqrt{x^2 - 9} = 0 \), which simplifies to \( x^2 - 9 = 0 \). Solve this equation to find: \ \ \( x^2 = 9 \). Thus, \( x = ±3 \). Hence, the vertical asymptotes are \( x = 3 \) and \( x = -3 \).

Step 2 - Identify the horizontal asymptote

Horizontal asymptotes can be found by analyzing the behavior of the function as \( x \) approaches infinity. For large values of \( x \), \( h(x) = \frac{x}{\sqrt{x^2 - 9}} \). Simplify this ratio by dividing numerator and denominator by \( x \): \ \ \( h(x) = \frac{x/x}{\sqrt{x^2 - 9}/x} = \frac{1}{\sqrt{1 - 9/x^2}} \). As \( x \) approaches infinity or negative infinity, \( \frac{9}{x^2} \) approaches zero, so \( h(x) \) approaches \( \frac{1}{\sqrt{1}} = 1 \). Therefore, the horizontal asymptote is \( y = 1 \).

Step 3 - Sketch the graph

To sketch the graph, note the vertical asymptotes at \( x = 3 \) and \( x = -3 \), and the horizontal asymptote at \( y = 1 \). Plot key points and draw the function approaching these asymptotes. For example: \( h(0) = - \frac{1}{3} \), etc.

Key Concepts

Vertical Asymptotes

A vertical asymptote in a graph is a line that the function approaches infinitely close, but never actually meets. It occurs where the denominator of a fraction is zero, creating a situation where the function heads towards infinity. To find vertical asymptotes for the function \( h(x) = \frac{x}{\sqrt{x^2 - 9}} \), we need to analyze the denominator:

\[ \sqrt{x^2 - 9} \]

This expression becomes zero when \( x^2 - 9 = 0 \). Solving for \( x \):

\[ x^2 = 9 \]
\[ x = \pm 3 \]

Therefore, the vertical asymptotes for this function are at \( x = 3 \) and \( x = -3 \). These lines, \( x = 3 \) and \( x = -3 \), will be boundaries that the curve will approach but never intersect.

Horizontal Asymptotes

A horizontal asymptote is a y-value that the function approaches as \( x \) heads towards positive or negative infinity. For the function:

\[ h(x) = \frac{x}{\sqrt{x^2 - 9}} \]

We determine the horizontal asymptote by examining the ratio of the leading terms in the numerator and the denominator as \( x \) grows large.
For large values of \( x \):

\[ h(x) \approx \frac{x}{\sqrt{x^2}} = \frac{x}{|x|} \], since \( \sqrt{x^2} = |x| \)

When \( x \) is positive, \( \frac{x}{|x|} = 1 \), and when \( x \) is negative, \( \frac{x}{|x|} = -1 \).

Here, as \( x \) approaches infinity, the horizontal asymptote is \( y = 1 \). Likewise, as \( x \) approaches negative infinity, it would approach \( y = -1 \).

Graphing Functions

When graphing the function \( h(x) = \frac{x}{\sqrt{x^2 - 9}} \), the key steps involve: identifying asymptotes, plotting key points, and sketching the curve based on these features. Follow these steps:

• Identify the vertical asymptotes at \( x = 3 \) and \( x = -3 \).


• Identify the horizontal asymptote, which from our discussion above, is \( y = 1 \).

After determining the asymptotes, choose some key values to plot. Example points include:

• \( h(0) = -\frac{1}{3} \)
• \( h(1) \approx 0.577 \)

Next, draw the vertical and horizontal asymptotes on your graph.
Plot the points and ensure the curve approaches these asymptotes. For instance, as \( x \) gets closer to 3 or -3, \( h(x) \) heads towards negative and positive infinity respectively.
Complete the graph with a consistent smooth curve, reflecting the behavior near the asymptotes.