Problem 65 - 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. $$f(x)=\frac{(x+4)^{2}}{(x-1)(x+5)}$$

Step-by-Step Solution

Answer

The graph has vertical asymptotes at \(x = 1\) and \(x = -5\), a horizontal asymptote at \(y = 1\), and touches the x-axis at \(x = -4\).

Step 1: Identify the Vertical Asymptotes

Vertical asymptotes occur at the values of \(x\) that make the denominator equal to zero, but not the numerator. Set the denominator equal to zero: \((x-1)(x+5) = 0 \). This gives us vertical asymptotes at \(x = 1\) and \(x = -5\).

Step 2: Identify the Horizontal Asymptote

For rational functions \(f(x) = \frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials, the horizontal asymptote depends on their degrees. Here, the degrees of both the numerator \((x+4)^2\), and the denominator \((x-1)(x+5)\) are 2. Thus, the horizontal asymptote is \(y = \frac{1}{1} = 1\).

Step 3: Find the Y-Intercept

To find the y-intercept, evaluate \(f(0)\). Substitute \(x = 0\) into the function: \(f(0) = \frac{(0+4)^2}{(0-1)(0+5)} = \frac{16}{-5} = -3.2\). The y-intercept is \((-3.2)\).

Step 4: Find the X-Intercepts

The x-intercepts occur where the numerator equals zero: \((x+4)^2 = 0 \). Solving this gives \(x = -4\), a double root, so \((-4, 0)\) is an x-intercept with multiplicity 2.

Step 5: Analyze Behavior Near Asymptotes and Intercepts

Near the vertical asymptotes \(x = 1\) and \(x = -5\), check intervals to determine the direction of the graph. As \(x\) approaches these points, \(f(x)\) will tend toward \(+\infty\) or \(-\infty\). At \(x = -4\), since it's a double root, the graph will touch and rebound off the x-axis.

Step 6: Sketch the Graph

Use the found asymptotes, intercepts, and the behavior analysis to sketch. Draw vertical dotted lines at \(x = 1\) and \(x = -5\), a horizontal dotted line at \(y = 1\), and plot the intercepts. Show the double root at \(x = -4\) causing the curve to bounce off the x-axis. Complete the sketch considering graphing symmetry and behavior around the asymptotes.

Key Concepts

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. In rational functions, these are crucial for understanding the graph's behavior.

• Vertical Asymptotes: These occur at the values of \(x\) that make the denominator zero, provided they do not cancel with zeros from the numerator. For the function \(f(x) = \frac{(x+4)^2}{(x-1)(x+5)}\), set the denominator equal to zero: \((x-1)(x+5) = 0\). Solving this gives vertical asymptotes at \(x = 1\) and \(x = -5\). These vertical lines, \(x = 1\) and \(x = -5\), indicate where the function heads off towards infinity.
• Horizontal Asymptote: The horizontal asymptote is determined by comparing the degrees of the numerator and the denominator. For this rational function, both are of degree 2. Thus, the horizontal asymptote is \(y = \frac{1}{1} = 1\). This horizontal line depicts the value that \(f(x)\) approaches as \(x\) tends toward infinity or negative infinity.

By identifying these asymptotes, you set boundaries on the graph sketch and understand the overall direction of the curve.

X-Intercepts

X-intercepts are points where the graph crosses the x-axis. These occur when the numerator of the rational function is zero. To determine the x-intercepts of the function \(f(x) = \frac{(x+4)^2}{(x-1)(x+5)}\):

• Set the numerator \((x+4)^2\) equal to zero. Solving \((x+4)^2 = 0\) gives you \(x = -4\).
• The solution \(x = -4\) represents a double root because \(x+4\) is squared. This means at \((-4, 0)\), the graph touches the x-axis but does not cross it, indicating a multiplicity of 2.

Understanding x-intercepts helps you identify where the function reaches a value of zero, aiding in the overall graph depiction.

Y-Intercept

The y-intercept is where the function crosses the y-axis. This occurs when \(x = 0\). Finding the y-intercept for a rational function involves substituting zero for \(x\) in the expression.For \(f(x) = \frac{(x+4)^2}{(x-1)(x+5)}\), calculate the y-intercept by evaluating \(f(0)\):

• Substitute \(x = 0\) in the function: \(f(0) = \frac{(0+4)^2}{(0-1)(0+5)} = \frac{16}{-5} = -3.2\).
• Thus, the y-intercept is at the point \((0, -3.2)\).

This point gives you a specific location on the graph and helps in accurately sketching the curve's placement relative to the axes.

Graph Sketching

Sketching the graph of a rational function involves assembling all the information you've gathered about intercepts and asymptotes, and analyzing function behavior near these key areas.

• First, draw vertical asymptotes as dotted lines at \(x = 1\) and \(x = -5\). These lines guide where the function shoots off to infinity.
• Plot the horizontal asymptote as a dotted line at \(y = 1\), showing where the curve levels off for extreme values of \(x\).
• Mark the x-intercept at \((-4, 0)\) and the y-intercept at \((0, -3.2)\).
• Near the double root \(x = -4\), the graph does not cross but bounces off the x-axis, reflecting the root's multiplicity.
• Analyze the function's direction towards its asymptotic lines as \(x\) approaches them, observing if it heads to \(+\infty\) or \(-\infty\).

By following these steps and carefully plotting, you can create an accurate sketch that represents the full behavior and shape of the rational function.