Problem 63 - College Algebra

Question

a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+x-6}{x-3}$$

Step-by-Step Solution

Answer

The slant asymptote for the rational function \(f(x)=\frac{x^{2}+x-6}{x-3}\) can be found using polynomial long division, which results in the linear equation \(y = x + 4\). Following the seven-step strategy further helps in graphing this function.

Step 1: Apply Polynomial Long Division

Firstly, divide the numerator \(x^{2} + x - 6\) by the denominator \(x - 3\) using polynomial long division. You could also use synthetic division as an alternative method.

Step 2: Find the Quotient

The quotient obtained from the division in step 1 gives the equation for the slant asymptote. Due to the degree of the numerator being exactly one more than the denominator, the division will yield a linear equation, indicating a slant asymptote, rather than a horizontal one.

Step 3: Identify Vertical Asymptotes

To start with the seven-step process, first identify the vertical asymptotes by setting the denominator equal to zero and solving for \(x\).

Step 4: Identify Horizontal Asymptotes

The next step of the process is to identify the horizontal asymptotes of the function. However, as identified before, this rational function has a slant asymptote, instead of a horizontal one, as the degree of the polynomial in the numerator is higher than the one in the denominator.

Step 5: Find the x and y-intercepts

In this step, you determine the x and y-intercepts for the function. The x-intercept is found by setting the numerator equal to zero and solving for \(x\). The y-intercept is found by substituting \(x=0\) into the function.

Step 6: Analyze the Sign and Plot Additional Points

Now, analyze the sign of the rational function in each interval determined by the vertical asymptotes. Also, plot additional points for a more accurate sketch of the function.

Step 7: Sketch the Graph

Finally, using all the information gathered, draw the graph of the function. Make sure to include the vertical asymptotes as dashed lines, the intercepts as dots, and show how the graph approaches the asymptotes.

Key Concepts

Polynomial Long Division

Before you can confidently graph a rational function, it's essential to understand polynomial long division. This process is reminiscent of long division with numbers, but instead, you're dividing polynomials.

Here's a simple flow: First, you take the lead term of the dividend (the numerator) and divide it by the lead term of the divisor (the denominator). Write the result above the division line. Then you multiply the entire divisor by that result and subtract it from the dividend. Repeat the steps until the degree of the remainder is less than the divisor. Why bother with this? Well, it helps you find the slant asymptote, which is simply the quotient without the remainder when the numerator's degree is one higher than the denominator's.

Slant Asymptote

A slant asymptote—also known as an oblique asymptote—occurs when the polynomial in the numerator is one degree higher than the polynomial in the denominator of a rational function. Unlike horizontal asymptotes that suggest what happens as x approaches infinity, slant asymptotes are linear and provide a diagonal line that the graph gets closer to as |x| becomes larger.

Once you have the result from polynomial long division, discard the remainder. The quotient is your straight-line equation for the slant asymptote. Graph this line on your coordinate plane with a dotted line to signify it's an asymptote, not a part of the function's graph.

Vertical Asymptote

Vertical asymptotes are vertical lines that the graph of a function approaches but never touches or crosses. They indicate places where the function is not defined—in other words, points of infinite discontinuity. To find them, set the denominator of your rational function to zero and solve for x.

The solutions are the x-values where vertical asymptotes occur. It’s important to remember that these lines are barriers that the graph can’t cross as they represent values that would make the denominator—and thus the function—undefined.

X-Intercepts and Y-Intercepts

Understanding intercepts is vital for graphing functions. The x-intercept(s) occur where the graph crosses the x-axis, indicating that y is zero at these points. To find them, set the numerator of the rational function to zero and solve for x.

The y-intercept is found where the graph crosses the y-axis, which happens when x is zero. To find it, substitute x with zero in the rational function and solve for y. These intercept points give you concrete spots through which the function will pass, anchoring the graph to specific locations on the coordinate plane.

Rational Function Analysis

Rational function analysis is the process of examining the features and behavior of a rational function. It typically involves finding asymptotes, intercepts, and analyzing the function's behavior around these lines and points.

You should also determine the function's end behavior: how does the function behave as x approaches infinity or negative infinity? Look at intervals of increase and decrease and identify any symmetries. By taking these steps, you build a comprehensive understanding of the function's graph which includes its shape, position, and other critical attributes.