Problem 3 - 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) $$f(x)=\frac{1}{x+12}$$

Step-by-Step Solution

Answer

The x-axis is its horizontal asymptote (Option G).

Step 1: Determine the x-intercept

To find the x-intercept, set the numerator of the rational function equal to zero and solve for x. For \(f(x) = \frac{1}{x+12}\), the numerator is 1, which never equals zero. Hence, there is no x-intercept.

Step 2: Determine the y-intercept

The y-intercept is found by evaluating the function at \(x = 0\). Substituting, \(f(0) = \frac{1}{0+12} = \frac{1}{12}\). Therefore, the y-intercept is \((0, \frac{1}{12})\), not \((0, 5)\).

Step 3: Determine the vertical asymptote

To find the vertical asymptote, set the denominator equal to zero and solve. For the function \(f(x) = \frac{1}{x+12}\), set \(x+12 = 0\), leading to \(x = -12\). Therefore, the vertical asymptote is \(x = -12\), not options D or H.

Step 4: Determine the horizontal or oblique asymptote

Compare the degrees of the numerator and the denominator. Here, the degree of the numerator (1) is less than the degree of the denominator (1), so the horizontal asymptote is the x-axis, \(y=0\). This matches option G.

Step 5: Check for a hole in the graph

A hole occurs when the numerator and denominator have common factors. Since \(f(x) = \frac{1}{x+12}\) does not have common factors in both the numerator and the denominator, there is no hole. Therefore, option E is incorrect.

Key Concepts

Asymptotes

Asymptotes play a crucial role in understanding the behavior of rational functions. They are lines that the graph of the function approaches but never touches. Usually, rational functions can have both vertical and horizontal asymptotes.Vertical asymptotes occur where the denominator of the rational function equals zero, creating a limit where the function tends to infinity. In the example function, \( f(x) = \frac{1}{x+12} \), setting the denominator \( x+12 \) to zero gives us the vertical asymptote at \( x = -12 \). This means the graph will approach this line but will never actually touch or cross it.Horizontal asymptotes indicate the behavior of the function as \( x \) approaches infinity or negative infinity. In our specific function, the degree of the numerator is less than the degree of the denominator, resulting in a horizontal asymptote at \( y = 0 \). The graph flattens out and approaches the x-axis as \( x \) goes to extreme values. Note that a function can cross a horizontal asymptote at finite points but won't in this specific case.

Intercepts

Intercepts are specific points where a graph crosses the axes. These are important as they provide points that are easily plotted, assisting in sketching the function's overall behavior.The **x-intercepts** are points where the graph crosses the x-axis. This happens where the numerator of the function equals zero. For \( f(x) = \frac{1}{x+12} \), since the numerator is 1, which never becomes zero, there are no x-intercepts. Thus, this function does not touch or cross the x-axis at any point other than potentially at infinity.The **y-intercept** is where the graph crosses the y-axis, found by evaluating the function at \( x = 0 \). At \( f(0) \), substituting zero gives us \( \frac{1}{12} \), indicating a y-intercept at \( (0, \frac{1}{12}) \). This tells us exactly where the function will hit the y-axis, providing a fundamental point for graphing.

Graphical Analysis

Graphical analysis involves studying these key characteristics to sketch and understand the behavior of a rational function completely.

• Vertical Asymptotes: Inform us of where the function will have undefined points and thus, sharp turns or breaks in the graph.
• Horizontal Asymptotes: Help predict the behavior of the graph as it extends towards infinity. It provides insight into the end behavior of the function.
• Intercepts: Provide key points that anchor the graph on the coordinate plane, which are essential for getting started with sketching the graph.
• Holes: Although not present in this particular function, these occur when the numerator and denominator share factors, resulting in undefined points that aren't asymptotes.

Learning the interplay between these features is essential for a strong understanding of rational functions. They allow one to picture how the graph behaves, transitions, and behaves asymptotically or even intermittently. Visualizing these intersections and asymptotes gives a clear guide to sketch these graphs accurately and comprehend their nature.