Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{-x}{x^{2}+x-6}$$

Step-by-Step Solution

Answer

The x-intercept is \(x=0\), the y-intercept is \(y=0\), there are no points of symmetry, the vertical asymptotes are \(x=2\) and \(x=-3\) and the horizontal asymptote is \(y=0\).

Step 1: Factor the Denominator

To start, we should factor the denominator \(x^{2}+x-6\), which can be factored into \((x-2)(x+3)\).

Step 2: Find the X and Y Intercepts

The x-intercepts are the values of x for which y is zero. This is when the numerator equals to 0. Hence, \(x=0\) is the x-intercept. The y-intercept is the value of y when x is 0. In this case, \(y=0\) is the y-intercept.

Step 3: Check for Symmetry

We check for symmetry by substituting \(-x\) into \(x\). If the new function equals the original function, it is symmetrical around the y-axis. If the new function equals the original function multiplied by \(-1\), it is symmetrical around the x-axis. In our case, neither holds true, so there is no symmetry.

Step 4: Identify Asymptotes

Vertical asymptotes occur when the denominator equals 0 and the numerator is not 0, which is \(x=2, -3\) for this function. The horizontal asymptote can be found by looking at the highest exponent of x. The coefficient of \(x^2\) in the denominator is 1, and there is no \(x^2\) in the numerator. Therefore, \(y=0\) is the horizontal asymptote.

Step 5: Sketch the Graph

Using the obtained details, we can sketch the graph of the function. The graph should cross the x-axis at \(x=0\) and the y-axis at \(y=0\). The graph must also approach but never cross the vertical asymptotes at \(x=2\) and \(x=-3\), and the horizontal asymptote at \(y=0\).

Key Concepts

Understanding Vertical Asymptotes

A vertical asymptote is a line where a rational function is undefined due to the denominator equalling zero. This occurs because division by zero is impossible. To find vertical asymptotes, set the denominator equal to zero and solve for x. In our function, the denominator is factored into

• (x - 2)
• (x + 3)

By solving these equations

• x - 2 = 0 resulting in x = 2
• x + 3 = 0 resulting in x = -3

we determine that the vertical asymptotes are at x = 2 and x = -3. As you approach these values from either side, the output of the function tends towards infinity or negative infinity, never actually touching or crossing the vertical line.

Discovering Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a function as the input becomes very large or very small. To determine them, compare the degrees of the polynomial in the numerator and denominator:

• Unequal Degrees: If the degree of the numerator is less than that of the denominator, the horizontal asymptote is y = 0.
• Same Degree: If the degrees are the same, divide the leading coefficients.
• Greater Numerator: If the numerator's degree is higher, there is no horizontal asymptote.

In this case, the numerator

• -x

is of degree 1, while the denominator

• x^2 + x - 6

is of degree 2. Therefore, this function has a horizontal asymptote at y = 0 because the numerator degree is less than that of the denominator.

Finding Intercepts

Intercepts are where the graph touches or crosses the axes:

• X-Intercepts: Occur when y = 0. Solve by setting the numerator to zero. In this function, -x = 0, so the x-intercept is at (0,0). The graph crosses the x-axis at this point.
• Y-Intercepts: Occur when x = 0. Plug x = 0 into the function: \( f(0) = \frac{-0}{0^2+0-6} = 0\), so the y-intercept is also at (0,0).

Intercepts provide specific points that help anchor the sketch of a graph.

Graph Sketching Basics

Graph sketching involves plotting the significant features of a function to understand its behavior visually. Start by noting the intercepts at (0,0), indicating where the graph crosses the axes. Next, identify vertical asymptotes at x = 2 and x = -3, where the graph will approach but never touch these lines, creating sharp lines where the graph veers off. Notice the horizontal asymptote at y = 0, showing that the function's values come close to zero as x moves towards positive or negative infinity. When sketching, aim for a gently curving line approaching the asymptotes without crossing them, reflecting the function's long-term behavior.