Problem 75 - 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. $$f(x)=\frac{3 x^{3}+2 x^{2}-12 x-8}{x^{2}+x+4} ;[-4.7,4.7] \text { by }[-3.1,3.1]$$

Step-by-Step Solution

Answer

(0, -2) for y-intercept; no x-intercepts. No vertical asymptotes. Oblique asymptote: \( y = 3x - 1 \). Domain: all reals; Range: all reals.

Step 1: Enter the Function into Calculator

First, input the function \( f(x) = \frac{3x^3 + 2x^2 - 12x - 8}{x^2 + x + 4} \) into a graphing calculator and set the window to \([-4.7, 4.7]\) for the x-axis and \([-3.1, 3.1]\) for the y-axis to visualize the graph over the specified range.

Step 2: Find the Intercepts

To find the x-intercept(s), set \( f(x) = 0 \). The function will equal zero when the numerator is zero. Solve \( 3x^3 + 2x^2 - 12x - 8 = 0 \) using the calculator. For the y-intercept, substitute \( x = 0 \) into the function: \( f(0) = \frac{-8}{4} = -2 \). Thus, the y-intercept is at \( (0, -2) \).

Step 3: Check for Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function is zero and the numerator is not zero. For \( f(x) \), solve \( x^2 + x + 4 = 0 \). This quadratic has no real roots (as the discriminant \( 1^2 - 4 \times 1 \times 4 = -15 \) is negative), so there are no vertical asymptotes.

Step 4: Determine the Oblique Asymptote

An oblique asymptote may exist if the degree of the numerator is exactly one greater than the degree of the denominator. Here, the degree of the numerator is 3 and the degree of the denominator is 2. Perform polynomial division of the numerator \( 3x^3 + 2x^2 - 12x - 8 \) by the denominator \( x^2 + x + 4 \) to find the asymptote equation. The result is \( y = 3x - 1 \).

Step 5: State the Domain and Range

The domain of the function is all real numbers except where the denominator equals zero. Since there are no real roots of \( x^2 + x + 4 \), the domain is all real numbers. For the range, the behavior of \( f(x) \) is influenced by the oblique asymptote \( y = 3x - 1 \), covering most real numbers except certain values approached asymptotically.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool when dealing with rational functions like the one given: \( f(x) = \frac{3x^3 + 2x^2 - 12x - 8}{x^2 + x + 4} \).By entering this function into the graphing calculator and setting the window to \([-4.7, 4.7]\) for the x-axis and \([-3.1, 3.1]\) for the y-axis, you gain a visual perspective of the function's behavior. The calculator helps to illustrate key features of the graph, such as intercepts and asymptotic behavior. This visual aid can be especially useful for identifying trends or patterns that may not be immediately obvious from the analytic form of the function alone.

Intercepts

Intercepts are the points where the graph of a function crosses the axes. To find the x-intercepts of the function \( f(x) = \frac{3x^3 + 2x^2 - 12x - 8}{x^2 + x + 4} \), set the numerator equal to zero and solve: \( 3x^3 + 2x^2 - 12x - 8 = 0 \).You can use the graphing calculator to find the roots of this equation, which represent the x-intercepts. For the y-intercept, substitute \( x = 0 \) into the function: \( f(0) = \frac{-8}{4} = -2 \). Thus, the y-intercept is at the point \( (0, -2) \). These intercepts provide crucial insights into where the function changes direction or crosses the axes.

Asymptotes

Asymptotes characterize the behavior of a function as it approaches certain points. For the given function, there are no vertical asymptotes. Vertical asymptotes appear where the denominator is zero but the numerator is not. Solving \( x^2 + x + 4 = 0 \), you find no real roots, thus no vertical lines where the function blows up. Instead, this function has an oblique asymptote, which occurs because the degree of the numerator (3) is one more than the degree of the denominator (2). Using polynomial division, the oblique asymptote is found to be \( y = 3x - 1 \). This equation tells you how the function behaves as \( x \) becomes very large or very small.

Domain and Range

Domain and range define the possible x and y values a function can take. For the function \( f(x) = \frac{3x^3 + 2x^2 - 12x - 8}{x^2 + x + 4} \), the domain is all real numbers, since there are no x-values causing the denominator to be zero.This allows the function to accept any real number input. The range, influenced by the oblique asymptote \( y = 3x - 1 \), encompasses most real numbers. As x approaches extremes, the function aligns closely with the asymptote, suggesting the absence of bounds on the y-values except those approaching asymptotically.