Problem 39 - A Graphical Approach to College Algebra

Question

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(x+6)(x-2)}{(x+3)(x-4)}$$

Step-by-Step Solution

Answer

Vertical asymptotes at \(x=-3\) and \(x=4\); horizontal asymptote at \(y=1\). Intercepts at \((-6,0)\), \((2,0)\), \((0,1)\).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is equal to zero, but the numerator is not zero. Set each factor of the denominator to zero and solve: \((x+3)=0\) gives \(x=-3\), and \((x-4)=0\) gives \(x=4\). Since these values do not make the numerator zero, \(x=-3\) and \(x=4\) are vertical asymptotes.

Step 2: Identify Horizontal Asymptote

The horizontal asymptote is determined by the degrees of the numerator and denominator. Both have degree 2. Therefore, the horizontal asymptote is \(y=\frac{1}{1}=1\), found by the ratio of the leading coefficients.

Step 3: Find the Intercepts

To find the x-intercepts, set the numerator equal to zero: \((x+6)(x-2)=0\). Solving gives \(x=-6\) and \(x=2\). For the y-intercept, set \(x=0\) in \(f(x)\), so \(f(0)=\frac{(0+6)(0-2)}{(0+3)(0-4)}=\frac{-12}{-12}=1\). Thus, the y-intercept is \((0,1)\).

Step 4: Sketch the Graph

Plot the intercepts \((-6,0)\), \((2,0)\), and \((0,1)\). Draw vertical dashed lines at \(x=-3\) and \(x=4\) for vertical asymptotes. Draw a horizontal dashed line at \(y=1\) for the horizontal asymptote. Use these guides to sketch the graph, showing how the function behaves near the intercepts and asymptotes.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are one of the key features to consider when graphing a rational function. They occur at the values of \(x\) that make the denominator zero, provided that these values do not also make the numerator zero. In our given function, \(f(x) = \frac{(x+6)(x-2)}{(x+3)(x-4)}\), the denominator becomes zero when \(x = -3\) and \(x = 4\). Because these values do not make the numerator zero (i.e., \((x+6)(x-2)eq 0\) at \(x = -3\) and \(x = 4\)), each is a vertical asymptote. Vertical asymptotes are represented by vertical dashed lines on the graph. As the graph approaches these lines, the function will increase or decrease without bound. It's important to understand that the graph can never actually touch or cross a vertical asymptote.

Horizontal Asymptote

Horizontal asymptotes help to determine the end behavior of a rational function as \(x\) heads towards positive or negative infinity. To find the horizontal asymptote, compare the degrees of the polynomial in the numerator and the polynomial in the denominator. In the given function, both the numerator \((x+6)(x-2)\) and the denominator \((x+3)(x-4)\) are quadratic polynomials, which means the degree is 2 for both. When the degrees are the same, the horizontal asymptote is y = the ratio of the leading coefficients of the numerator and the denominator. For the given function, both the leading coefficients are 1, giving us a horizontal asymptote at \(y = \frac{1}{1} = 1\). This asymptote is represented by a horizontal dashed line on the graph. As \(x\) becomes very large or very small, the graph will approach this line, indicating the direction the function values settle into.

Intercepts in Graphs

Intercepts are the points where the graph intersects with the axes. For rational functions, finding these intercepts provides specific points that can guide the graphing process.To find the x-intercepts, set the numerator equal to zero, because points on the x-axis have a y-value of zero. With \(f(x) = \frac{(x+6)(x-2)}{(x+3)(x-4)}\), the x-intercepts are found by solving \((x+6)(x-2)=0\), leading to \(x = -6\) and \(x = 2\). These points, \((-6, 0)\) and \((2, 0)\), will be marked on the x-axis.The y-intercept occurs where the graph crosses the y-axis, which happens when \(x = 0\). Substitute \(x = 0\) into the function: \(f(0) = \frac{(0+6)(0-2)}{(0+3)(0-4)} = \frac{-12}{-12} = 1\). Therefore, the y-intercept is \((0, 1)\). These intercepts provide crucial anchor points when sketching the function's graph and show where the graph enters and exits the quadrants. Keep in mind that understanding how a function intersects the axes is vital for an accurate sketch.