Problem 27 - Algebra 2

Question

Sketch the graph of each rational function. $$ y=\frac{x+4}{x-4} $$

Step-by-Step Solution

Answer

The function graph \( y = \frac{x+4}{x-4} \) has a vertical asymptote at \(x=4\), a horizontal asymptote at \(y=1\), and x and y intercepts at \(x=-4\) and \(y=-1\) respectively. The curve of the function approaches but never crosses the asymptotes.

Step 1: Identify the Vertical Asymptote

Set the denominator \(x - 4\) equal to zero and solve for x, which yields \(x = 4\). This is the vertical asymptote of the rational function.

Step 2: Find the Horizontal Asymptote

Since the degrees of the numerator and denominator of the rational function are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 1 and the leading coefficient of the denominator is also 1, hence the horizontal asymptote is \(y = 1\).

Step 3: Find the x and y Intercepts

To find the x-intercept, set \(y = 0\) and solve for \(x\), which yields \(x = -4\). For the y-intercept, set \(x = 0\) and solve for \(y\), yielding \(y = -1\). These are the x and y intercepts respectively.

Step 4: Sketch the Graph

Sketch the graph of the function using the information obtained from the previous steps. Draw horizontal and vertical asymptotes at \(x = 4\) and \(y = 1\). Mark the x-intercept at -4, the y-intercept at -1. Then sketch the curve of the function which approaches but never crosses the asymptotes.

Key Concepts

Vertical Asymptote

When dealing with rational functions, vertical asymptotes are an essential feature to understand. A vertical asymptote occurs at values of x where the function is undefined. Essentially, this is when the denominator of the rational function equals zero, implying that the function's value approaches infinity or negative infinity.

To find the vertical asymptote for the function \( y = \frac{x+4}{x-4} \), we set the denominator \( x-4 \) equal to zero:

• \( x-4 = 0 \) leads to \( x = 4 \).

This indicates a vertical asymptote at \( x = 4 \), meaning the graph of the function will get indefinitely close to the line \( x = 4 \) but will never actually touch or cross it.

Remember, vertical asymptotes reflect values where the function is not defined, making them crucial for shaping the behavior of the graph.

Horizontal Asymptote

Horizontal asymptotes in rational functions provide insights into the behavior of the function as \( x \) approaches positive or negative infinity. They show that, far out along the x-axis, the function gets closer and closer to the line defined by the horizontal asymptote.

To determine the horizontal asymptote of a rational function, compare the degrees of the numerator and the denominator:

• If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is less, the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator is greater, there is no horizontal asymptote.

In this case, both the numerator and the denominator of \( y = \frac{x+4}{x-4} \) have a degree of 1, thus:

• Leading coefficients are both 1, so the horizontal asymptote is \( y = \frac{1}{1} = 1 \).

This tells us that as \( x \) moves toward infinity in either direction, the function will approach the line \( y = 1 \).

X-Intercept

The x-intercept of a function represents the point where the graph crosses the x-axis, meaning the output value \( y \) is zero. To find the x-intercept of a rational function:

• Set the numerator equal to zero and solve for \( x \).

For \( y = \frac{x+4}{x-4} \), the x-intercept occurs where:

• \( x+4 = 0 \), which simplifies to \( x = -4 \).

Thus, the graph of the function crosses the x-axis at the point \( (-4, 0) \).

Notice that this value does not affect the denominator; hence, the function is defined and the x-intercept exists where expected.

Y-Intercept

The y-intercept of a function is the point where the graph crosses the y-axis, occurring when \( x \) is zero. To find the y-intercept of a rational function, set \( x \) to zero and solve for \( y \).

For \( y = \frac{x+4}{x-4} \), substituting \( x = 0 \) gives:

• \( y = \frac{0+4}{0-4} = \frac{4}{-4} = -1 \).

Therefore, the y-intercept of the function is at the point (0, -1).

This indicates that when the graph touches the y-axis, it does so at this particular point, providing a starting point to sketch the curve along with other critical features.