Problem 22 - Precalculus with Calculus Previews

Question

Find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{x(x-5)}{x^{2}-9} $$

Step-by-Step Solution

Answer

Vertical asymptotes: \( x = 3, x = -3 \). Horizontal asymptote: \( y = 1 \). Intercepts: \( (0, 0), (5, 0) \).

Step 1: Identify Vertical Asymptotes

To find the vertical asymptotes of the function \( f(x)=\frac{x(x-5)}{x^{2}-9} \), we set the denominator equal to zero and solve for \( x \). The denominator is \( x^{2}-9 = 0 \), so \( x^{2} = 9 \). Solving for \( x \) gives \( x = 3 \) and \( x = -3 \). Therefore, the vertical asymptotes are \( x = 3 \) and \( x = -3 \).

Step 2: Determine Horizontal Asymptotes

The degrees of the numerator and denominator are both 2, leading us to compare their leading coefficients. The leading term in the numerator is \( x^2 \) and the leading term in the denominator is \( x^2 \). Comparing these, we see the coefficients are both 1, so the horizontal asymptote is \( y = \frac{1}{1} = 1 \).

Step 3: Find x-Intercepts

Set the numerator equal to zero and solve for \( x \). The equation from the numerator is \( x(x-5) = 0 \), which means \( x = 0 \) and \( x = 5 \) are the \( x \)-intercepts.

Step 4: Calculate y-Intercept

Find the \( y \)-intercept by evaluating \( f(0) \). Substituting 0 into \( f(x) \), we have \( f(0) = \frac{0(0-5)}{0^2-9} = 0 \). Thus, the \( y \)-intercept is \( (0, 0) \).

Step 5: Sketch the Graph

To sketch the graph, mark the vertical asymptotes at \( x = -3 \) and \( x = 3 \), draw the horizontal asymptote along \( y = 1 \), and plot the intercepts \( (0, 0) \) and \( (5, 0) \). The function behaves asymptotically near the vertical lines and approaches \( y = 1 \) as \( x \to \pm\infty \).

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, leading the function to approach infinity. In other words, these vertical lines are where the graph splits and the function is undefined. For the function \( f(x)=\frac{x(x-5)}{x^{2}-9} \), finding vertical asymptotes involves setting the denominator \( x^{2} - 9 \) equal to zero.

• Start by solving the equation \( x^{2} - 9 = 0 \).
• This simplifies to \( x^{2} = 9 \).
• Finally, solve for \( x \) to get \( x = 3 \) and \( x = -3 \).

So, the function has vertical asymptotes at \( x = 3 \) and \( x = -3 \). These asymptotes will appear as vertical lines on the graph, indicating where the function cannot be evaluated.

Horizontal Asymptotes

Horizontal asymptotes provide insight into how a function behaves as \( x \) moves towards positive or negative infinity. For rational functions, they are determined by comparing the degrees of the numerator and the denominator.

For \( f(x)=\frac{x(x-5)}{x^{2}-9} \):

• The degree of the numerator \( x(x-5) \) is 2—resulting from \( x \times x \).
• The degree of the denominator \( x^{2} - 9 \) is also 2.

When these degrees are equal, the horizontal asymptote can be found by dividing the leading coefficients of the numerator and denominator.

• Both leading coefficients are 1, resulting in a horizontal asymptote of \( y = \frac{1}{1} = 1 \).

Thus, the graph will tend towards \( y = 1 \) as \( x \) approaches infinity in either direction.

X-Intercepts

The \( x \)-intercepts of a graph are the points where the graph crosses the \( x \)-axis, meaning the output is zero. To find the \( x \)-intercepts for the rational function \( f(x)=\frac{x(x-5)}{x^{2}-9} \), set the numerator equal to zero.

• The equation \( x(x-5) = 0 \) simplifies to two solutions: \( x = 0 \) and \( x = 5 \).

Therefore, the \( x \)-intercepts are at the points \( (0, 0) \) and \( (5, 0) \), where the graph intersects the \( x \)-axis. These intercepts are important as they anchor the curve of the function onto the \( x \)-axis.

Y-Intercepts

The \( y \)-intercepts of a graph show where the graph crosses the \( y \)-axis. This happens when \( x = 0 \). To calculate the \( y \)-intercept of \( f(x)=\frac{x(x-5)}{x^{2}-9} \), substitute \( x = 0 \) into the function.

• Evaluate \( f(0) = \frac{0(0-5)}{0^{2}-9} \).
• This simplifies to \( f(0) = 0 \).

Thus, the \( y \)-intercept is \( (0, 0) \). The point where the graph touches the \( y \)-axis is instrumental in determining the graph's initial behavior.