Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{x+4}{x-5}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{x+4}{x-5}\) includes intercepts at \((-4,0)\) and \((0,-\frac{4}{5})\). It has no symmetry, a vertical asymptote at \(x=5\), and a horizontal asymptote at \(y=1\).

Step 1: Find the Intercepts

To find the x-intercept, set \(y=f(x)=0\) and solve for \(x\). Similarly, for y-intercept, set \(x=0\). Solving for x, we get \(x=-4\). For y, we get \(y=-\frac{4}{5}\). So, the intercepts are \((-4, 0)\) and \((0, -\frac{4}{5})\).

Step 2: Check for Symmetry

A graph is symmetric about the y-axis if replacing \(x\) by \(-x\)produces the same equation. If replacing \(x\) by \(-x\) produces an equivalent equation except with all terms having the opposite sign, it's symmetric about the origin. Here, replacing \(x\) by \(-x\), we get \(-\frac{x+4}{x+5}\) which is not equivalent to the original function. Thus, the function is not symmetric.

Step 3: Find the Vertical Asymptotes

The vertical asymptote of a rational function are the values of \(x\) where the denominator equals zero. Therefore, solving the denominator \(x-5=0\), we get \(x=5\). So, \(x=5\) is the vertical asymptote.

Step 4: Find the Horizontal Asymptotes

Horizontal asymptotes can be found by looking at the degrees of the polynomials in the numerator and denominator. Here, the degrees of both the numerator and denominator are 1 and the coefficients of the highest degree term are 1. The horizontal asymptote is therefore \(y=\frac{1}{1}=1\).

Key Concepts

Understanding Intercepts in Rational Functions

Intercepts are key points where a graph crosses the axes. They tell us significant information about the behavior of a function. For rational functions like \( f(x) = \frac{x+4}{x-5} \), finding intercepts is straightforward.

• To find the **x-intercept**, set the entire function equal to zero: \( f(x) = 0 \). This means solving the numerator for zero, which gives \( x = -4 \). The x-intercept is at \( (-4, 0) \).
• For the **y-intercept**, substitute \( x = 0 \) into the function. This gives \( y = -\frac{4}{5} \). So, the y-intercept is at \( (0, -\frac{4}{5}) \).

Intercepts are a starting point for sketching the graph of the function. They confirm where the function crosses the x and y axes, providing clues about the function's direction and curvature. Remember, intercepts are simple but powerful tools in graphing.

Exploring Vertical Asymptotes in Rational Functions

Vertical asymptotes in a rational function indicate where the function is undefined due to division by zero. Let's look closely at \( f(x) = \frac{x+4}{x-5} \).

• Identify the denominator and set it equal to zero: \( x-5=0 \).
• Solve this equation to find \( x = 5 \).

Therefore, \( x = 5 \) is the vertical asymptote. As \( x \) approaches 5, the function values increase or decrease without bound, showcasing the function's undefined behavior at this point. Vertical asymptotes act like invisible walls the graph cannot touch or cross. Note that at \( x = 5 \), the graph veers off towards infinity on either side, which can dramatically affect the graph's appearance.

Discovering Horizontal Asymptotes in Rational Functions

Horizontal asymptotes represent the value that a function approaches as \( x \) goes to positive or negative infinity. They provide insights into the long-term behavior of the function.For \( f(x) = \frac{x+4}{x-5} \), the process is simple:

• Examine the degrees of the polynomials in both the numerator and denominator. Here, both are degree 1.
• The horizontal asymptote is found by taking the ratio of the leading coefficients. In this function, that ratio is \( \frac{1}{1} = 1 \).

Thus, \( y = 1 \) is the horizontal asymptote.Horizontal asymptotes hint at the ceiling (or floor) of a graph as it stretches towards infinity. The function will hover around this y-value in the far ends of the x-axis, but may touch or cross the line closer to the origin. Remember, it's more about what happens at the extremes!