Problem 85 - College Algebra

Question

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 of the function \(f(x)=\frac{x^{2}+x-6}{x-3}\) is the line \(y = x+4\). The graph approaches this line for large values of \(|x|\), passes through the intercepts \(x = -3, 2\) and \(y=-2\), and also follows the asymptotic behavior towards the vertical asymptote at \(x=3\).

Step 1: Simplify the function

The given function is \(f(x)=\frac{x^{2}+x-6}{x-3}\). This can be simplified using division as follows: \( f(x) = x + 4 + \frac{6}{x-3} \)

Step 2: Find the vertical asymptote

This can be found by setting the denominator equal to zero and solving for x. So, \(x-3=0\), which simplifies to \(x=3\). So, the vertical asymptote is \(x=3\).

Step 3: Find the horizontal asymptote

Since the degree of the numerator is one more than the degree of the denominator, there is no horizontal asymptote of this rational function.

Step 4: Determine the slant asymptote

The slant asymptote is actually the quotient when the function is divided. Therefore, in our case, the line \(y = x+4\) is the slant asymptote.

Step 5: Determine the x and y intercepts

Set \(f(x)=0\) to find the x-intercepts: \(0=\frac{x^{2}+x-6}{x-3}.\) After we simplify, we find that the x-intercepts are \(x = -3\) and \(x = 2\). The y-intercept can be found by setting \(x=0: f(0)=\frac{0^{2}+0-6}{0-3}=-2\), which simplifies to \(y=-2\).

Step 6: Plot the asymptotes and intercepts

Plot the vertical asymptote \(x=3\), the slant asymptote \(y = x+4\), and the intercepts \(x = -3, 2\) and \(y=-2\). This will give a skeletal graph which will help in drawing the rational function.

Step 7: Draw the graph using the plotted points

Following the asymptotic behavior, the graph approaches the asymptotes, and passes through the intercepts and known other points such as \(f(4)=2, f(-1)=1\).

Key Concepts

Rational Functions

Rational functions are a type of function represented as a ratio of two polynomials. The general form of a rational function is \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)eq 0\). These functions are defined for all real numbers except where the denominator equals zero, which leads to an undefined value.

A key characteristic of rational functions is their potential for asymptotic behavior as they increase or decrease without bounds. As mentioned in the original exercise, the given function is \(f(x)=\frac{x^{2}+x-6}{x-3}\). To simplify this rational function, we can divide the numerator by the denominator, which helps in identifying the asymptotes and determining the graph's behavior.

Vertical Asymptote

A vertical asymptote occurs in a rational function at values of \(x\) that make the denominator zero, as these are points where the function is undefined. In simple terms, a vertical asymptote is a vertical line that the graph of the function approaches but never touches or crosses.

In the given exercise, to find the vertical asymptote, solve \(x - 3 = 0\). This calculation tells us that there is a vertical asymptote at \(x = 3\). When graphing, remember that the function will not exist exactly on the vertical asymptote line, and the graph will tend to go towards infinity or negative infinity as it approaches these lines.

Graphing Techniques

Graphing rational functions involves a combination of identifying intercepts, asymptotes, and understanding where the function is defined. In the step-by-step solution, several lessons are displayed on optimal graphing techniques:

• Find the x-intercepts by solving \(f(x)=0\).
• Determine the y-intercept by evaluating the function at \(x=0\).
• Identify the vertical and slant asymptotes as the guides for sketching the graph.

Start by plotting these known values on the graph. Once you have the intercepts and asymptotes, sketch the general shape of the curve, ensuring it reflects the asymptotic behavior, approaching but never crossing the vertical asymptotes.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as its inputs get very large or very small. In rational functions, asymptotes play a critical role in this behavior. As we observe the function near the asymptotes, it "hugs" them, showing its inclination to follow closely without actually meeting.

In this exercise, the slant asymptote is \(y = x + 4\), revealing that as \(x\) becomes very large or small, the function \(f(x)\) behaves similarly to the line \(y = x + 4\), deviating only by the fraction that reduces as \(x\) increases. This behavior assists in graph sketching, as you know that at extreme ends, the graph will draw close to the slant line, and vertically it approaches infinity as it nears its vertical asymptotes.