Problem 4 - A Graphical Approach to College Algebra

Question

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. (Column II) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote. (Column I) $$f(x)=\frac{-3+x^{2}}{x^{2}}$$

Step-by-Step Solution

Answer

The vertical asymptote is the y-axis.

Step 1: Identify the Rational Function Type

The given function is \( f(x) = \frac{-3 + x^2}{x^2} \). Since the degree of the numerator and the denominator are the same (both are 2), this is a rational function that might have a horizontal asymptote.

Step 2: Identify Horizontal Asymptote

Since the degrees of the numerator and denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients. Here, both coefficients are equal to 1, so the horizontal asymptote is \( y = \frac{1}{1} = 1 \). This does not match any options in Column II.

Step 3: Identify Vertical Asymptotes and Holes

To find vertical asymptotes, set the denominator equal to zero: \( x^2 = 0 \) gives \( x = 0 \). This suggests a possible vertical asymptote at \( x = 0 \). However, since \( x = 0 \) gives a non-zero result in the numerator, it confirms \( x = 0 \) as a vertical asymptote, matching with option H.

Step 4: Identify x-intercepts

Set the numerator equal to zero: \( -3 + x^2 = 0 \) results in \( x^2 = 3 \), so \( x = \pm \, \sqrt{3} \). This provides two x-intercepts, none of which match \((-3,0)\).

Step 5: Identify y-intercepts

Substitute \( x = 0 \) into the function: \( f(0) = \frac{-3 + 0}{0} \) is undefined due to division by zero, so there is no y-intercept.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes in rational functions occur when we look at the behavior of the function as the value of \(x\) approaches infinity or negative infinity. These asymptotes provide information about the function's end behavior; that is, they tell us how the function behaves as \(x\) gets very large or very small.

To find the horizontal asymptote of a rational function, consider the degrees of the numerator and the denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y=0\).
• If the degrees are equal, as in the rational function \(f(x) = \frac{-3+x^2}{x^2}\), the horizontal asymptote is \(y\) equal to the ratio of the leading coefficients. Here, this would be \(y=\frac{1}{1}=1\).
• If the degree of the numerator is greater than the denominator, there is no horizontal asymptote, although there may be an oblique one.

Identifying the horizontal asymptote helps sketch out the graph, giving insights into how it flattens out. Knowing this, you can see that for our function, \(y=1\) doesn't match any description from the exercise list.

Vertical Asymptotes

Vertical asymptotes occur where a rational function's denominator is equal to zero, and the function goes to infinity. They represent values of \(x\) that the function cannot take on freely and usually indicate a division by zero that isn't canceled out by the numerator. These asymptotes typically manifest as a vertical line on a graph.

To identify vertical asymptotes in a rational function:

• Set the denominator equal to zero and solve for \(x\).
• If this results in a zero value in the denominator but a non-zero numerator, a vertical asymptote exists.

In the case of \(f(x) = \frac{-3+x^2}{x^2}\), setting the denominator equal to zero gives \(x^2=0\), which means \(x=0\). Here, the numerator does not lead to zero, confirming there is a vertical asymptote at \(x=0\), corresponding to option H in the exercise. Recognizing vertical asymptotes helps us understand the key restrictions on the function and what happens near those values of \(x\).

X-intercepts

X-intercepts are the points where the graph of a function crosses the \(x\)-axis. These points occur where the function's value is zero, meaning that the numerator of the rational function must equal zero since dividing zero by any non-zero number equals zero.

To find x-intercepts:

• Set the numerator equal to zero and solve for \(x\).

For \(f(x) = \frac{-3+x^2}{x^2}\), setting \(-3+x^2 = 0\) gives \(x^2=3\), resulting in \(x = \pm \sqrt{3}\). This function has x-intercepts at these points, though neither equals \(-3\), as none coincide with \((-3,0)\) from the exercise. Knowing the x-intercepts allows us to plot crucial intersections on the graph, giving further insight into the function's shape and behavior.