Problem 93 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to graph 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. $$f(x)=\frac{3 x^{3}+2 x^{2}-12 x-8}{x^{2}+x+4} ;[-6.6,6.6] \text { by }[-4.1,4.1]$$

Step-by-Step Solution

Answer

x-intercepts found from roots of numerator; y-intercept is (0, -2). No vertical asymptotes due to denominator having no real roots; oblique asymptote is \( y = 3x - 1 \). Domain and range are all real numbers.

Step 1: Graph the Function

Use a graphing calculator or graphing software to plot the rational function \( f(x) = \frac{3x^3 + 2x^2 - 12x - 8}{x^2 + x + 4} \) in the window \([-6.6, 6.6]\) by \([-4.1, 4.1]\). Observe the shape of the graph and any intercepts or asymptotic behavior.

Step 2: Find x- and y-intercepts

To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):\( 3x^3 + 2x^2 - 12x - 8 = 0 \).Using a calculator, we find the approximate roots (x-intercepts) of the cubic equation by graphing or numerical methods.For the y-intercept, set \( x = 0 \) in \( f(x) \):\( f(0) = \frac{-8}{4} = -2 \).Thus, the y-intercept is at \( (0, -2) \).

Step 3: Explain the Absence of Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is non-zero at those values. For \( x^2 + x + 4 = 0 \), calculate the discriminant \( b^2 - 4ac = 1^2 - 4 \times 1 \times 4 = -15 \). Since the discriminant is negative, there are no real solutions to the denominator being zero, thus no vertical asymptotes.

Step 4: Find the Equation of the Oblique Asymptote

Since the degree of the numerator is one more than the degree of the denominator, perform polynomial long division on \( \frac{3x^3 + 2x^2 - 12x - 8}{x^2 + x + 4} \). The quotient gives the oblique asymptote: \( 3x - 1 \). Thus, the equation of the oblique asymptote is \( y = 3x - 1 \).

Step 5: Determine the Domain and Range

The domain of \( f(x) \) is all real numbers, \( \mathbb{R} \), since the denominator has no real roots (no values make it zero). For the range, observe the behavior of the graph and note that due to the oblique asymptote, as \( x \rightarrow \pm \infty \), \( y \rightarrow \pm \infty \). Therefore, the range is approximately all real numbers, \( \mathbb{R} \).

Key Concepts

Graphing Calculators

Graphing calculators are incredibly valuable tools when it comes to visualizing functions, especially rational functions like the one given in this exercise. By inputting the function \( f(x) = \frac{3x^3 + 2x^2 - 12x - 8}{x^2 + x + 4} \) into a graphing calculator, you can easily observe its behavior over a specific interval.

• Graphing allows you to see the overall shape of the function.
• It can highlight key features such as intercepts and asymptotic behavior.

When plotting this function in the window \([-6.6, 6.6]\) by \([-4.1, 4.1]\), you will notice the general curvature of the graph. This visualization helps in understanding how it behaves across its domain. By using a graphing calculator, complex calculations are simplified, making intercepts and asymptotes easier to identify. It can save considerable time and ensure accuracy, especially when dealing with intricate polynomial equations.

Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. In the context of rational functions, there are usually two types of asymptotes: vertical and oblique (or slant) asymptotes.

Vertical Asymptotes

• These occur when the denominator of the rational function approaches zero, as long as the numerator doesn’t also become zero at the same values.

In the given function, the denominator \( x^2 + x + 4 \) never equals zero, as evidenced by its negative discriminant (-15). This means there are no real solutions where the denominator would be zero, resulting in no vertical asymptotes.

Oblique Asymptotes

For oblique asymptotes, they occur when the degree of the numerator is exactly one more than the degree of the denominator. In this function, the numerator is a cubic polynomial, while the denominator is quadratic.To find the equation of the oblique asymptote, we perform polynomial long division between the numerator and the denominator, resulting in the linear equation \( y = 3x - 1 \). This line is approached by the graph at extreme values of \( x \).

Polynomial Long Division

Polynomial long division is an essential skill for manipulating rational functions, especially when determining oblique asymptotes. It works similarly to long division of numbers but involves variable expressions. In the given function \( \frac{3x^3 + 2x^2 - 12x - 8}{x^2 + x + 4} \), the degree of the numerator is one more than the degree of the denominator, signaling the presence of an oblique asymptote.

• Begin by dividing the leading term of the numerator by the leading term of the denominator.
• Multiply the entire divisor by this quotient.
• Subtract the result from the original numerator.
• Repeat this process with the new polynomial that results after each subtraction until the degree of the polynomial remainder is less than the degree of the divisor.

For this function, after performing the division, the quotient \( 3x - 1 \) represents the oblique asymptote, demonstrating a critical concept in graphing rational functions.

Intercepts

Intercepts are crucial for understanding the position and orientation of a graph in the coordinate plane. They indicate where the graph crosses the axes.

X-intercepts

The x-intercepts of a function are found by setting the function equal to zero and solving for \( x \). For our rational function, \( 3x^3 + 2x^2 - 12x - 8 = 0 \), graphing or numerical methods reveal the approximate locations of the intercepts. These intercepts provide points where the graph crosses the x-axis.

Y-Intercept

To find the y-intercept, set \( x = 0 \) in the function and solve for \( y \). In this case, \( f(0) = \frac{-8}{4} = -2 \), indicating the y-intercept at the point \( (0, -2) \).Intercepts used alongside other features like asymptotes offer a fuller picture of the function’s behavior. Identifying intercepts is straightforward but fundamental in analyzing and graphing rational functions.