Problem 77 - 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. $$f(x)=\frac{x^{3}+4 x^{2}-x-4}{-2 x^{2}-2 x-4} ;[-4.7,4.7] \text { by }[-3.1,3.1]$$

Step-by-Step Solution

Answer

x-intercepts cannot be pinpointed exactly without graph; y-intercept is (0, 1); no vertical asymptotes; oblique asymptote is \( y = -\frac{1}{2}x - 1 \); domain is all real numbers; range is all real numbers.

Step 1: Input Function into Graphing Calculator

Enter the rational function \( f(x) = \frac{x^{3} + 4x^{2} - x - 4}{-2x^{2} - 2x - 4} \) into your graphing calculator. Set the viewing window to \([-4.7, 4.7]\) for the x-axis and \([-3.1, 3.1]\) for the y-axis. This will help visualize the graph of the function over the specified range.

Step 2: Find x-Intercepts by Setting Numerator to Zero

To find the x-intercepts, set the numerator \( x^{3} + 4x^{2} - x - 4 \) equal to zero and solve for \( x \). Use graphing technology or algebraic methods to find the roots of the cubic equation. Check the graph to confirm the x-intercepts.

Step 3: Determine the y-Intercept

To find the y-intercept, substitute \( x = 0 \) into the function: \[ f(0) = \frac{0^{3} + 4 \cdot 0^{2} - 0 - 4}{-2 \cdot 0^{2} - 2 \cdot 0 - 4} = \frac{-4}{-4} = 1 \]Thus, the y-intercept is at (0, 1).

Step 4: Analyze for Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero and causes the function to be undefined. Solve \(-2x^{2} - 2x - 4 = 0\) for \( x \). However, the numerator will also make this zero at these points, indicating removable discontinuities rather than vertical asymptotes. Calculate the common factors and verify with graph.

Step 5: Determine the Oblique Asymptote

To find the oblique asymptote, perform polynomial long division of the numerator by the denominator. The quotient will provide the equation for the oblique asymptote. In this case:\( x^{3}+4x^{2}-x-4 \) divided by \(-2x^{2}-2x-4\) yields the quotient \( -\frac{1}{2}x - 1 \), which is the oblique asymptote.

Step 6: Determine the Domain and Range

For the domain, exclude values that result in zero for the denominator (after confirming these don't create vertical asymptotes). The range is typically all real numbers unless restricted. Here, the domain is all real numbers, but check for any plot discontinuities. The range, confirmed by the graph, is also all real numbers due to the oblique asymptote.

Key Concepts

x-intercepts

To find the x-intercepts of a rational function like \( f(x) = \frac{x^3 + 4x^2 - x - 4}{-2x^2 - 2x - 4} \), you need to focus on the numerator. The x-intercepts occur where the numerator equals zero. This is because, for any fraction, the whole expression will equal zero when the numerator is zero (provided the denominator is not zero at the same time).
\[ x^3 + 4x^2 - x - 4 = 0 \]
Solving this cubic equation provides the x-intercepts. These can be computed with algebraic techniques or confirmed with a graphing calculator.

• Identify potential rational roots using the Rational Root Theorem.
• Use synthetic division or another method to confirm the roots.

Check these roots on the graph to ensure they match the x-intercepts. These points will cross or touch the x-axis, showing where the function value is zero.

y-intercepts

The y-intercept of a function is the point where the graph crosses the y-axis. To find this, substitute \( x = 0 \) into the function. For the given function \( f(x) = \frac{x^3 + 4x^2 - x - 4}{-2x^2 - 2x - 4} \), replace \( x \) with zero:
\[ f(0) = \frac{0^3 + 4 \cdot 0^2 - 0 - 4}{-2 \cdot 0^2 - 2 \cdot 0 - 4} = \frac{-4}{-4} = 1 \]
This calculation shows that the y-intercept is at the point \((0, 1)\).

• At the y-intercept, the graph will touch or cross the y-axis.
• This point is easy to verify on a graph.

This gives you valuable insight into how the function behaves near the origin.

oblique asymptote

An oblique asymptote appears when the degree of the numerator is exactly one more than the degree of the denominator. For the rational function \( f(x) = \frac{x^3 + 4x^2 - x - 4}{-2x^2 - 2x - 4} \), the numerator is of degree 3 and the denominator is of degree 2. This difference of one signals the presence of an oblique asymptote.
To find it, perform polynomial long division of the numerator by the denominator. The quotient (ignoring the remainder) is the equation of the oblique asymptote:
\[-\frac{1}{2}x - 1\]

• The graph approaches this line as \( x \) becomes very large or very small.
• Typically, the oblique asymptote represents the end behavior of the graph.

Thus, understanding this asymptote helps describe how the graph behaves at extreme values.

domain and range

The domain and range of a rational function are fundamental aspects to consider. The domain represents all possible input values (\( x \)) that don't lead to an undefined function — typically values that do not zero out the denominator. For \( f(x) = \frac{x^3 + 4x^2 - x - 4}{-2x^2 - 2x - 4} \),
find where the denominator equals zero:
\[-2x^2 - 2x - 4 = 0\]
Both the numerator and denominator equal zero at these points, suggesting removable discontinuities rather than true vertical asymptotes.
Despite this, these points are excluded from the domain.
The domain is all real numbers except the roots of this equation.

• Graphing visually checks domain restrictions.
• Use algebraic solutions for precise calculations.

The range is generally all real numbers here; however, check the graph for any restrictions. As it shows an oblique asymptote, the range is not constrained to specific values.