Problem 78 - A Graphical Approach to College Algebra

Question

Use a calculator to graph each rational function in the window indicated. Then (a) give the \(x\) - and y-intercepts, (b) explain why there are no vertical asymptotes, (c) give the equation of the oblique asymptote, and (d) give the domain and range. $$f(x)=\frac{-x^{3}-7 x^{2}+16 x+112}{x^{2}+x+28} ;[-15,10] \text { by }[-5,15]$$

Step-by-Step Solution

Answer

The x-intercepts are approximately (-8, 0) and (4, 0), y-intercept at (0, 4). No vertical asymptotes, oblique asymptote is \(y = -x - 8\), domain and range are all real numbers.

Step 1: Graph the Rational Function

Use a calculator or graphing tool to plot the function \( f(x)=\frac{-x^{3}-7 x^{2}+16 x+112}{x^{2}+x+28} \) within the window \([-15,10] \times [-5,15]\). Analyze the curve to find important points and features for subsequent steps.

Step 2: Find the x- and y-intercepts

Determine the x-intercepts by solving the numerator \(-x^3-7x^2+16x+112=0\). The y-intercept occurs when \(x = 0\); substitute x = 0 into the function to find \(f(0)\). The approximate intercepts are the following, based on graphical output: x-intercept(s) at approximately (-8,0) and (4,0); y-intercept at (0,4).

Step 3: Check for Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero, if not canceled by the numerator. Set \(x^2 + x + 28 = 0\) and solve for \(x\). The roots are complex (discriminant is negative), so there are no vertical asymptotes.

Step 4: Determine the Oblique Asymptote

For an oblique asymptote, divide the numerator by the denominator. Perform polynomial long division: dividing \(-x^{3}-7x^{2}+16x+112\) by \(x^{2}+x+28\) gives a quotient of \(-x-8\). Thus, the oblique asymptote is \(y = -x - 8\).

Step 5: Define the Domain and Range

Domain: All real numbers except where the denominator is zero, but since roots are complex, the domain is all real numbers. Range: Based on the graph behavior and oblique asymptote, the range includes all real numbers due to the polynomial degree dominance in horizontal behavior.

Key Concepts

x- and y-intercepts

Understanding the intercepts of a rational function is key to graphing and analyzing the function's behavior. The

• **x-intercepts** occur where the graph of the function crosses the x-axis. For this, the value of the function must be zero, meaning we only focus on the numerator. In the given function, \(-x^3-7x^2+16x+112=0,\)solving it approximately yields x-intercepts at (-8,0) and (4,0). These are the points where the graph intersects the x-axis.
• **y-intercepts** occur where the graph crosses the y-axis. This happens when the value of x is zero. Simply substitute 0 for x in the function. Evaluating the ratio gives \(f(0) = \frac{112}{28} = 4\),resulting in the y-intercept at (0,4).

Graphs of the function typically confirm these intercepts as visible crossing points on the axes. Being aware of where these intercepts lie helps in sketching the behavior of the function correctly.

Vertical and Oblique Asymptotes

Asymptotes represent lines that a graph approaches but never truly reaches. **Vertical asymptotes** occur when the denominator of a rational function is zero, creating undefined points unless the numerator can cancel these zeros. In \(f(x) = \frac{-x^{3}-7x^{2}+16x+112}{x^{2}+x+28},\)set \(x^{2} + x + 28 = 0.\)Here, attempts to find real roots result in complex solutions, meaning no real vertical asymptotes exist. This implies the graph continues unbroken in these regions. **Oblique asymptotes** arise when the degree of a polynomial in the numerator is exactly one more than the degree of the denominator. Performing polynomial division of \(-x^{3} -7x^{2} +16x +112\) by \(x^{2}+x+28\) yields \(-x-8,\)identifying an oblique asymptote at \(y = -x - 8.\)The graph gets closer and closer to this line indefinitely as x moves towards positive or negative infinity.

Domain and Range

The domain and range describe all possible values input and output by the function.

• The **domain** of a rational function usually excludes values that create division by zero. Nevertheless, the denominator \(x^2 + x + 28\) lacks real roots, resulting in a domain of all real numbers \(-\infty < x < \infty\).
• The **range** involves analyzing the function's output behavior as x varies. The influence of an oblique asymptote \(y = -x - 8\) showcases general dominance of the polynomial's leading terms as x tends to infinities, implying the range also covers all real numbers \(-\infty < y < \infty\).

The broad domain and range demonstrate that the function extends without boundary over both axes, forming a complete view of its continuous nature.