Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{3 x}{x^{2}-x-2}$$

Step-by-Step Solution

Answer

Firstly, we associate x and y intercepts. Then we factor the function to find the vertical asymptotes and holes by finding values of x that make the denominator zero. As the degree of numerator is less than the degree of denominator, the horizontal asymptote is \(y=0\). With all these, we are informed to draw the function. Lastly, we can confirm our sketch with a graphing tool.

Step 1: Find the x and y intercepts

To find the x-intercept, we need to solve \(f(x) = 0\), or \(\frac{3x}{x^2 -x -2} = 0\). Solving this will give the x-intercepts. For the y-intercept, substitute x=0 in the function to find \(f(0)\).

Step 2: Identify Vertical Asymptotes and Holes

To identify vertical asymptotes and holes, we first simplify the equation and see where the denominator equals to zero. Vertical asymptotes exist where the denominator becomes zero while the rational function is not. A hole is the root of the denominator which is also root of numerator. So we factor the equation as \(f(x) = \frac{3x}{(x-2)(x+1)}\). The vertical asymptotes are defined by \(x-2=0\) and \(x+1=0\) without being a hole (root for both).

Step 3: Determine the Horizontal Asymptotes

Horizontal asymptotes are determined by the degree of the equation. As the degree of numerator is less than the degree of denominator, the function approaches 0 as x approaches \(+\infty\) and \(-\infty\). Hence \(y=0\) is the horizontal asymptote.

Step 4: Draw the Graph

Having obtained all the necessary points, asymptotes, and holes, we can now sketch the graph. Remember, the function approaches vertical asymptotes from both directions and horizontal asymptotes as \(x \rightarrow +\infty\) and \(x \rightarrow -\infty\).

Step 5: Verify with a graphing tool

Once we have sketched the function by hand, we should use a graphing tool to verify that the sketch matches the actual graph.

Key Concepts

Vertical Asymptotes

A vertical asymptote in a rational function is where the function heads towards infinity, either positively or negatively, as the x-value approaches a particular point. To identify vertical asymptotes in a rational function like \(f(x) = \frac{3x}{x^2 - x - 2}\), you first look at the denominator.

• You factor the denominator into \((x-2)(x+1)\).
• Then, determine where each factor equals zero. In this case, \(x-2 = 0\) and \(x+1 = 0\) gives potential vertical asymptotes at \(x = 2\) and \(x = -1\).

However, you must ensure these points are not holes, which occur when both the numerator and denominator have a common factor. In this example, since the numerator doesn't have these roots, both points are true vertical asymptotes. These asymptotes indicate that the function will sharply rise or fall without bound as it nears these x-values.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as \(x\) moves towards positive or negative infinity. For a rational function like \(f(x) = \frac{3x}{x^2 - x - 2}\), you determine horizontal asymptotes by comparing the degrees of the numerator and the denominator.

• If the degree of the numerator is less than the degree of the denominator, like here where the numerator is degree 1 (\(3x\)) and the denominator is degree 2 (\(x^2-x-2\)), the horizontal asymptote is \(y=0\).
• This means the function approaches the x-axis as \(x\) moves towards both \(+\infty\) and \(-\infty\).

Thus, you can expect the graph to level out to \(y = 0\) on both sides as \(x\) becomes very large or very small.

Graphing Rational Functions

When graphing rational functions, several characteristics need to be evaluated, such as asymptotes, intercepts, and holes. Let's walk through graphing \(f(x) = \frac{3x}{x^2 - x - 2}\).

• First, identify intercepts where the graph will intersect the axes. The x-intercept is found by setting the numerator to zero: \(3x = 0\) so \(x = 0\). The y-intercept is found by plugging \(x = 0\) into the function, yielding \(f(0) = 0\), so the intercepts are at \((0,0)\).
• Next, locate the vertical asymptotes at \(x = 2\) and \(x = -1\), and the horizontal asymptote at \(y = 0\).
• There are no holes in this function as the roots of the denominator are not in the numerator.

After marking these details, sketch the function. Start from known intercepts and follow approaching lines towards the asymptotes. Check the behavior as \(x\) approaches these critical areas, showing that the graph tends to infinity along vertical asymptotes and flattens towards the horizontal asymptote. Once you sketch by hand, use graphing utilities to verify your sketch accurately represents the actual function curve.