Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x-5}{x+3}$$

Step-by-Step Solution

Answer

The graph has vertical asymptote at \( x = -3 \), horizontal asymptote at \( y = 1 \), x-intercept at 5, and y-intercept at \(-\frac{5}{3}\).

Step 1: Identify Horizontal Asymptote

To find the horizontal asymptote of the function \( f(x) = \frac{x-5}{x+3} \), we look at the degrees of the polynomials in the numerator and denominator. Both the numerator and denominator have degree 1. When degrees are equal, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients. Thus, the horizontal asymptote is \( y = \frac{1}{1} = 1 \).

Step 2: Identify Vertical Asymptote

Vertical asymptotes occur where the denominator of the function is zero and the numerator is not zero. Set the denominator equal to zero: \( x + 3 = 0 \). Solving gives \( x = -3 \). Therefore, there is a vertical asymptote at \( x = -3 \).

Step 3: Find X-Intercept

The x-intercept of a function is found by setting the numerator equal to zero: \( x - 5 = 0 \). Solving this gives \( x = 5 \). Therefore, the graph intersects the x-axis at \( (5, 0) \).

Step 4: Find Y-Intercept

The y-intercept of a function is found by evaluating \( f(0) \). Substitute \( x = 0 \) into the function: \( f(0) = \frac{0 - 5}{0 + 3} = -\frac{5}{3} \). Thus, the y-intercept is at \( (0, -\frac{5}{3}) \).

Step 5: Sketch the Graph

Using the identified intercepts and asymptotes, we can sketch the graph. We have: - Vertical asymptote at \( x = -3 \).- Horizontal asymptote at \( y = 1 \).- X-intercept at \( (5, 0) \).- Y-intercept at \( (0, -\frac{5}{3}) \).The function will approach the horizontal asymptote \( y = 1 \) as \( x \to \infty \) or \( x \to -\infty \), and the curve will get closer to the vertical asymptote as \( x \to -3^+ \) or \( x \to -3^- \), but never touch it.

Key Concepts

Understanding Asymptotes in Rational Functions

Asymptotes are lines that a graph approaches but never actually reaches. In the context of rational functions like \( f(x) = \frac{x-5}{x+3} \), these provide important structural features for the graph.

There are two primary types of asymptotes you'll encounter:

• Horizontal Asymptotes: These occur when the values of \( f(x) \) approach a fixed number as \( x \) becomes very large (both positive and negative). For our function, the horizontal asymptote is determined by comparing the degrees of the polynomials in the numerator and the denominator. If they are equal, as in this function, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients. Here, both leading coefficients are 1, so the horizontal asymptote is \( y = 1 \).
• Vertical Asymptotes: These occur at values of \( x \) that make the denominator zero, assuming that the numerator is non-zero at these points. This is where the function value becomes infinite, causing the graph to go indefinitely upward or downward. For our function, setting \( x + 3 = 0 \) gives us the vertical asymptote at \( x = -3 \).

Visualizing these asymptotes on a graph helps in understanding the behavior of the function as it approaches these lines. Knowing where the graph bends or turns to avoid "touching" the asymptotes is crucial for accurate graphing.

Determining Intercepts in Rational Functions

Intercepts are points where the graph crosses the axes. They serve as checkpoints when sketching the graph.

For rational functions like \( f(x) = \frac{x-5}{x+3} \), you'll examine two types of intercepts:

• X-Intercepts: Found by setting the numerator equal to zero. This is because the value of the function \( f(x) \) is zero anywhere it crosses the x-axis. For this exercise, solving \( x - 5 = 0 \) gives us the x-intercept at the point \( (5, 0) \).
• Y-Intercepts: Found by evaluating the function at \( x = 0 \). We're finding where the graph crosses the y-axis. Substituting \( x = 0 \) into the function, you get \( f(0) = -\frac{5}{3} \), giving the y-intercept at \( (0, -\frac{5}{3}) \).

These intercepts assist in anchoring the graph and provide perspective on how the graph interacts with the axes, adding essential points to guide the overall shape of the graph.

Graphing Rational Functions Effectively

Graphing rational functions like \( f(x) = \frac{x-5}{x+3} \) involves connecting all identified components - asymptotes and intercepts - for an accurate representation.

Here’s how you can sketch the graph step by step:

• Start by drawing the vertical and horizontal asymptotes on your graph. These are not parts of the graph itself, but lines your graph should approach closely.
• Plot the x-intercept and y-intercept identified. These points should lie on the graph precisely, so mark them clearly.
• With these guides in place, sketch the graph, tracing how it approaches the asymptotes. The graph will curve towards the horizontal asymptote, \( y = 1 \), as \( x \to \infty \) or \( x \to -\infty \).
• Near the vertical asymptote, \( x = -3 \), draw the curve such that it heads sharply up or down, depending on the sign of the function as it nears \( -3 \).

Ensuring that the graph never actually touches any asymptotes, and accurately passes through intercepts, is crucial for a precise graphical representation. Follow these steps for a clearer, more accurate understanding of rational functions' behavior.