Problem 8 - A Graphical Approach to Precalculus with Limits

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. \(\mathbf{I}\) \(f(x)=\frac{x+3}{x-6}\) \(\mathbf{II}\) 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.

Step-by-Step Solution

Answer

Match: A with \((-3,0)\) and G with horizontal asymptote \(y=1\). Rest do not match.

Step 1: Determine the Vertical Asymptote

A vertical asymptote occurs where the denominator of the rational function equals zero, given that the numerator is not zero at that point. For \( f(x) = \frac{x+3}{x-6} \), set the denominator \( x-6 = 0 \) which gives \( x = 6 \). Therefore, the vertical asymptote is \( x = 6 \). None of the options in Column II match \( x = 6 \), so we disregard vertical asymptote choices in Column II.

Step 2: Determine the Horizontal Asymptote

For rational functions where the degrees of the numerator and denominator are the same, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients. Here, both the numerator and denominator have degree 1, and the leading coefficients are 1. Thus, the horizontal asymptote is \( y = 1 \). Therefore, the horizontal asymptote matches option G.

Step 3: Find the x-intercept

To find the \( x \)-intercept, set \( f(x) = 0 \). This occurs when the numerator \( x+3 = 0 \), hence \( x = -3 \). The \( x \)-intercept is \((-3,0)\), which matches option A in Column II.

Step 4: Find the y-intercept

The \( y \)-intercept occurs when \( x = 0 \). For the function \( f(x) = \frac{x+3}{x-6} \), \( f(0) = \frac{0+3}{0-6} = -\frac{1}{2} \). Therefore, the \( y \)-intercept is \( (0, -\frac{1}{2}) \). None of the listed options match \((0, -\frac{1}{2})\), so disregard \( y \)-intercept choices in Column II.

Key Concepts

Vertical Asymptote

A vertical asymptote is a line where the graph of a rational function tends to infinity. It's important because it indicates a value that the function cannot take and where the function behaves erratically.

To find the vertical asymptote of a rational function such as \( f(x) = \frac{x+3}{x-6} \), set the denominator equal to zero and solve for \( x \).

* Step 1: \( x - 6 = 0 \)
* Step 2: Solve for \( x \), yielding \( x = 6 \).

The vertical asymptote is hence \( x = 6 \). On the graph, you'll notice that as \( x \) approaches 6, the function values become very large or very small, never actually touching the line \( x = 6 \).

Horizontal Asymptote

Horizontal asymptotes describe the end behavior of the function as \( x \) approaches infinity or negative infinity. For rational functions, the degree of the numerator and the degree of the denominator play crucial roles.

In this case, for \( f(x) = \frac{x+3}{x-6} \), both the numerator and denominator are of degree 1. This implies that the horizontal asymptote is obtained by dividing the leading coefficients. So,

* leading coefficient of numerator (\(x+3\)) = 1
* leading coefficient of denominator (\(x-6\)) = 1

Thus, the horizontal asymptote is \( y = \frac{1}{1} = 1 \).

As \( x \) becomes very large or very small, the value of \( f(x) \) approaches 1. This means that further out on the graph, it will appear to very closely approach \( y = 1 \), but not necessarily touch it.

Intercepts

Intercepts provide key points on a graph where it crosses the axes.

**Finding the x-intercept:**
For the rational function \( f(x) = \frac{x+3}{x-6} \), set \( f(x) = 0 \) to find where the graph crosses the x-axis.

* Step 1: Set the numerator \( x+3 = 0 \)
* Step 2: Solve for \( x \), yielding \( x = -3 \).

Thus, the x-intercept is \( (-3, 0) \).

**Finding the y-intercept:**
To find where the graph crosses the y-axis, evaluate \( f(x) \) at \( x = 0 \).

* \( f(0) = \frac{3}{-6} = -\frac{1}{2} \)

Consequently, the y-intercept is \( (0, -\frac{1}{2}) \). It is where the graph will touch the y-axis.

Graphical Analysis

Graphical analysis is an essential tool for understanding behavior patterns in rational functions. By combining information on asymptotes and intercepts, you can sketch an accurate graph.

**Steps in Graphical Analysis:**

• Vertical Asymptote: Place a dashed line at \( x = 6 \). The graph will approach but never touch this line.
• Horizontal Asymptote: Draw a dashed line at \( y = 1 \). This controls the function's end behavior.
• Intercepts: Plot \( x \)-intercept at \( (-3, 0) \) and \( y \)-intercept at \( (0, -\frac{1}{2}) \).

By analyzing how these elements interact, one can see the full picture of the function's behavior. The function will approach both asymptotes without touching them and will intersect the graph at intercept points, revealing its basic shape.