Problem 95 - 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. $$f(x)=\frac{x^{3}+4 x^{2}-x-4}{-2 x^{2}-2 x-4} ;[-6.6,6.6] \text { by }[-4.1,4.1]$$

Step-by-Step Solution

Answer

X-intercepts: real roots of numerator; Y-intercept: (0,1); No vertical asymptote; Oblique asymptote: y = -0.5x - 1; Domain: all real; Range: all real.

Step 1: Generating the Graph

First, input the function \(f(x) = \frac{x^3 + 4x^2 - x - 4}{-2x^2 - 2x - 4}\) into your graphing calculator. Set the viewing window to \([-6.6, 6.6]\) for the x-axis and \([-4.1, 4.1]\) for the y-axis. Generate the graph.

Step 2: Finding the Intercepts

To find the x-intercepts, set the numerator equal to zero: \(x^3 + 4x^2 - x - 4 = 0\) and solve for \(x\). The x-intercepts are the values where the graph crosses the x-axis. Similarly, the y-intercept is found by evaluating \(f(0)\). Calculate \(f(0) = \frac{0^3 + 4(0)^2 - 0 - 4}{-2(0)^2 - 2(0) - 4} = 1\). So, the y-intercept is \(1\).

Step 3: Vertical Asymptotes Analysis

For vertical asymptotes, set the denominator to zero: \(-2x^2 - 2x - 4 = 0\). Solving gives \(-2(x^2 + x + 2) = 0\), which is a quadratic equation with no real solutions as the discriminant \(b^2 - 4ac = 1 - 8 = -7\) is negative. No vertical asymptotes exist.

Step 4: Determining the Oblique Asymptote

To find the oblique asymptote, perform polynomial long division of \(x^3 + 4x^2 - x - 4\) by \(-2x^2 - 2x - 4\). The quotient represents the oblique asymptote, which is \(-0.5x - 1\). Thus, the oblique asymptote's equation is \(y = -0.5x - 1\).

Step 5: Defining Domain and Range

The domain includes all real numbers except where the function is undefined. Since there are no vertical asymptotes, the domain is all real numbers. The range is all real numbers because the graph continues indefinitely in the y-direction, affected by the oblique asymptote.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool for visualizing rational functions, especially when dealing with complex expressions. To begin, enter the rational function, such as \( f(x) = \frac{x^3 + 4x^2 - x - 4}{-2x^2 - 2x - 4} \), into your calculator. Adjust the window settings to appropriately view this function, using the given limits \([-6.6, 6.6]\) for the x-axis and \([-4.1, 4.1]\) for the y-axis. These settings ensure a full view of the function's behavior, revealing critical aspects like intercepts and asymptotes. A graphing calculator helps by translating equations into visual paths, making abstract concepts tangible.

X-Intercepts

Finding x-intercepts of a rational function involves setting the numerator equal to zero. For our function, solve the equation: \( x^3 + 4x^2 - x - 4 = 0 \). The solutions to this equation are the x-intercepts, which are the points where the function crosses the x-axis. Locating these intercepts is crucial, as they indicate where the function touches or crosses the horizontal axis. By identifying these points, you gain insights into the behavior and characteristics of the function.

Y-Intercepts

The y-intercept of a rational function is the point where the graph crosses the y-axis. This occurs when \( x = 0 \). To find the y-intercept for the function \( f(x) = \frac{x^3 + 4x^2 - x - 4}{-2x^2 - 2x - 4} \), substitute 0 for \( x \):

• The calculation leads to \( f(0) = \frac{0 - 4}{-4} = 1 \).
• Therefore, the y-intercept is \( (0, 1) \).

This point is important for understanding the starting value of the function when graphed.

Oblique Asymptotes

Oblique asymptotes occur in rational functions where the degree of the numerator is one higher than the degree of the denominator. To find the oblique asymptote of \( f(x) = \frac{x^3 + 4x^2 - x - 4}{-2x^2 - 2x - 4} \), perform polynomial long division.

• The result is \( y = -0.5x - 1 \), which is the equation for the oblique asymptote.
• This line gives insight into how the function behaves at extreme x-values, as the graph will approximate this line but never actually meet it.

Vertical Asymptotes

To determine vertical asymptotes, set the denominator of the rational function equal to zero and solve for \( x \). For the function at hand,

• \(-2x^2 - 2x - 4 = 0\)
• Solving this reveals no real solutions, as the discriminant is negative: \( 1 - 8 = -7 \).
• This means that there are no vertical asymptotes in this case.

Vertical asymptotes typically indicate where the function's value approaches infinity, thus knowing their absence is just as informative as their presence.

Domain and Range

The domain of a rational function is the set of all possible x-values, except where the denominator equals zero. Since

• there are no vertical asymptotes in \( f(x) = \frac{x^3 + 4x^2 - x - 4}{-2x^2 - 2x - 4} \),
• the domain is all real numbers.

The range describes all possible y-values. For this function, the range is also all real numbers, given that the graph is unbounded and influenced by the oblique asymptote. Knowing both domain and range is key to understanding the full scope of the function's behavior and where it "lives" in the coordinate plane.