Problem 39 - 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. $$f(x)=\frac{x+1}{x-4}$$

Step-by-Step Solution

Answer

The graph of \( f(x) = \frac{x+1}{x-4} \) has a vertical asymptote at \( x = 4 \), a horizontal asymptote at \( y = 1 \), x-intercept at \( x = -1 \),\ and y-intercept at \( y = -\frac{1}{4} \).

Step 1: Determine Vertical Asymptotes

A vertical asymptote occurs where the denominator is zero and the numerator is not zero. Set the denominator equal to zero: \( x - 4 = 0 \). Solving for \( x \) gives \( x = 4 \). Thus, the vertical asymptote is at \( x = 4 \).

Step 2: Determine Horizontal Asymptotes

For rational functions, if the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients.The leading coefficients of \( x + 1 \) and \( x - 4 \) are both 1, thus the horizontal asymptote is at \( y = \frac{1}{1} = 1 \).

Step 3: Find the Intercepts

To find the x-intercept, set \( f(x) = 0 \) and solve for \( x \): \( \frac{x+1}{x-4} = 0 \) implies \( x+1 = 0 \), therefore, \( x = -1 \).To find the y-intercept, set \( x = 0 \), which gives \( f(0) = \frac{0+1}{0-4} = -\frac{1}{4} \).Thus, the x-intercept is at \( (-1, 0) \) and the y-intercept is at \( (0, -\frac{1}{4}) \).

Step 4: Sketch the Graph

Draw the vertical asymptote as a dashed vertical line at \( x = 4 \) and the horizontal asymptote as a dashed line at \( y = 1 \). Plot the intercepts \( (-1,0) \) and \( (0, -\frac{1}{4}) \). Recognize that the graph approaches these asymptotes but never intersects them (apart from intercept points where applicable). Note the function's behavior around \( x = 4 \) to show that as \( x \to 4^+ \), \( f(x) \to +\infty \) and as \( x \to 4^- \), \( f(x) \to -\infty \). Draw the curve that approaches the asymptotes as \( x \) moves towards infinity or negative infinity.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are one of the key features in graphing rational functions. They occur at values of x that make the denominator zero, while the numerator remains non-zero. In our example function, \( f(x) = \frac{x+1}{x-4} \), the denominator is \( x-4 \). We set \( x-4 = 0 \) to find where the denominator becomes zero. This simplifies to \( x = 4 \). Therefore, there is a vertical asymptote at \( x = 4 \).Unlike other vertical lines on the graph, vertical asymptotes are not part of the function’s graph. Instead, they represent a boundary where the function's value increases or decreases without limit. This means that as \( x \) approaches 4 from the left (\( x \to 4^- \)), the function's value decreases towards negative infinity. Conversely, as \( x \) approaches 4 from the right (\( x \to 4^+ \)), the function's value increases towards positive infinity. When sketching the graph, draw this vertical asymptote as a dashed line at \( x = 4 \). By understanding this boundary behavior, you can predict how the graph will look around the asymptote.

Horizontal Asymptotes

Horizontal asymptotes are another critical aspect of rational functions, representing the value that the function approaches as \( x \) moves towards positive or negative infinity. For the function \( f(x) = \frac{x+1}{x-4} \), both the numerator and the denominator have the same degree, which is 1. When the degrees are equal, the horizontal asymptote is found by dividing the leading coefficients of the numerator and the denominator.In our function, the leading coefficient of both the numerator (\( x+1 \)) and the denominator (\( x-4 \)) is 1. Dividing these gives \( \frac{1}{1} = 1 \). Therefore, the horizontal asymptote is at \( y = 1 \). This asymptote indicates that as \( x \) becomes very large or very small, the value of \( f(x) \) gets closer to 1 but never actually reaches it. When plotting the graph, represent the horizontal asymptote with a dashed line at \( y = 1 \). Over large values of \( x \), the graph will appear to "flatten out" towards this line, demonstrating how horizontal asymptotes guide the end behavior of the function.

Graphing Rational Functions

Graphing rational functions involves plotting various elements, such as vertical and horizontal asymptotes, intercepts, and behavior around critical points. Let's explore how this applies to our specific function, \( f(x) = \frac{x+1}{x-4} \).First, identify and draw both vertical and horizontal asymptotes:

• Vertical asymptote at \( x = 4 \)
• Horizontal asymptote at \( y = 1 \)

Use dashed lines to indicate these asymptotes, showing where the function will approach but never actually crosses, except at potential intercepts.Next, find intercepts to locate key points on the graph:

• X-intercept: Occurs where \( f(x) = 0 \), or \( x+1 = 0 \rightarrow x = -1 \). So, the point is \((-1, 0)\).
• Y-intercept: Occurs where \( x = 0 \), or \( f(0) = -\frac{1}{4} \). Therefore, the point is \((0, -\frac{1}{4})\).

Plot these points on your graph to provide reference as you sketch the curve.When sketching the actual curve, notice how it behaves near the asymptotes. As \( x \to 4 \), the function exhibits vertical asymptotic behavior: from the left approaching negative infinity and from the right approaching positive infinity. Additionally, as \( x \to \pm \infty \), the curve will approach the horizontal asymptote at \( y = 1 \). By systematically combining these elements, you create a complete visual representation of how the rational function behaves across its domain.