Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes in the graph of the function. Then use a graphing utility to verify your results. $$f(x)=\frac{3 x^{2}-8 x+4}{2 x^{2}-3 x-2}$$

Step-by-Step Solution

Answer

The function \(f(x) = \frac{3x^2 - 8x + 4}{2x^2 - 3x - 2}\) has vertical asymptotes at \(x = -0.5\) and \(x = 2\), a horizontal asymptote at \(y = \frac{3}{2}\), no slant asymptotes and no holes.

Step 1: Determine Vertical Asymptotes

To find the vertical asymptotes of the given function, set the denominator equal to zero and solve for x. So solving the equation \(2x^2 -3x - 2 = 0\) gives solutions \(x = -0.5\) and \(x = 2\). There are no common solutions in the numerator and the denominator, hence the function has vertical asymptotes at \(x = -0.5\) and \(x = 2\).

Step 2: Determine Horizontal Asymptotes

A horizontal asymptote is determined by comparing the degrees of the polynomials in the numerator and the denominator. If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. Here the degree of the numerator and the denominator are both 2, thus the horizontal asymptote of the function is \(y = \frac{3}{2}\) (the ratio of the leading coefficients).

Step 3: Determine Slant Asymptotes

A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. In this case, the degrees of the polynomials in the numerator and the denominator are the same, so the function does not have a slant asymptote.

Step 4: Determine Any Holes

A hole in the function would exist if a particular x-value makes both the numerator and the denominator equal to zero. However, there are no such x-values for this function, so there are no holes.

Step 5: Use a Graphing Utility to Verify

This step involves using a graphing calculator or other software to graph the function and confirm the asymptotes and any holes visually. The graph should have vertical asymptotes at \(x = -0.5\) and \(x = 2\), and a horizontal asymptote at \(y = \frac{3}{2}\), with no holes.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are straight lines to which a graph of a function approaches but never touches as the independent variable either increases or decreases without bound. They indicate where a function is undefined, and they occur at values of x that make the denominator of a rational function equal to zero, provided they don't cancel out with similar factors in the numerator.

To identify these asymptotes in the given function, \( f(x)=\frac{3x^{2}-8x+4}{2x^{2}-3x-2} \), we set the denominator equal to zero and solve for x. Doing so, we find the vertical asymptotes at \( x = -0.5 \) and \( x = 2 \), since these x-values are not solutions in the numerator.

Horizontal Asymptotes

Horizontal asymptotes represent the behavior of a graph as the independent variable approaches positive or negative infinity. To determine the existence of a horizontal asymptote, we compare the degrees of the polynomials in the numerator and the denominator of the rational function.

If the degrees are equal, there is a horizontal asymptote at \( y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} \). For the function \( f(x) \), we find the degrees are equal at 2, indicating a horizontal asymptote at \( y = \frac{3}{2} \).

Slant Asymptotes

Slant asymptotes, also known as oblique asymptotes, occur when the degree of the polynomial in the numerator of a rational function is exactly one greater than the degree of the polynomial in the denominator. In such cases, the function may have a line that the graph approaches but does not cross as x becomes increasingly positive or negative.

However, for the function in our example \( f(x) \), since the degrees are the same, there is no slant asymptote. It's important to note this type of asymptote only applies when we have this specific degree relationship between the numerator and denominator.

Graphing Rational Functions

Graphing rational functions involves a combination of understanding asymptotes, intercepts, and the behavior of the function at extreme values of x. The steps typically include finding vertical and horizontal asymptotes, slant asymptotes if applicable, and any holes in the graph. Holes occur when a factor is canceled out in both the numerator and the denominator.

In our given function \( f(x) \), after determining the vertical and horizontal asymptotes, we graph the function using a tool or graphing calculator to visualize the behavior near these asymptotes. We would expect the graph to approach the asymptotes \( x = -0.5 \) and \( x = 2 \) vertically, and \( y = \frac{3}{2} \) horizontally, without crossing them. Since there are no common factors in the numerator and denominator that could cancel out, we conclude there are no holes in the graph, assuring that our graph accurately represents the function's behavior across all values of x.