Problem 6 - 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. \(\mathbf{I}\) \(f(x)=\frac{4 x+3}{x-7}\) \(\mathbf{II}\) 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

The function matches choice C: Horizontal asymptote is \(y=4\).

Step 1: Identify the x-intercept

To find the x-intercept of the function \(f(x) = \frac{4x+3}{x-7}\), set \(f(x) = 0\). This implies the numerator should be zero: \(4x + 3 = 0\). Solving for \(x\), we get \(x = -\frac{3}{4}\). Thus, the x-intercept is \((-\frac{3}{4}, 0)\). This doesn't match any choice for the x-intercept.

Step 2: Identify the y-intercept

To find the y-intercept, set \(x = 0\) in \(f(x) = \frac{4x+3}{x-7}\). This gives \(f(0) = \frac{4(0) + 3}{0 - 7} = \frac{3}{-7} = -\frac{3}{7}\). Thus, the y-intercept is \((0, -\frac{3}{7})\). This doesn't match any choice for the y-intercept.

Step 3: Determine the vertical asymptote

The vertical asymptote occurs where the denominator is zero. For \(x-7 = 0\), solve \(x = 7\). Therefore, the vertical asymptote is at \(x = 7\). This doesn't match any choice for the vertical asymptote.

Step 4: Determine the horizontal asymptote

For the horizontal asymptote, compare the degrees of the numerator and denominator. Both numerator and denominator have degree 1. The horizontal asymptote is \(y = \frac{4}{1} = 4\). Therefore, the horizontal asymptote is \(y = 4\) (choice C).

Step 5: Evaluate for potential holes

There is a hole if there are common factors in the numerator and denominator. The function \(\frac{4x+3}{x-7}\) has no common factors between the numerator and denominator. Therefore, there is no hole in the function.

Key Concepts

X-intercept

The concept of the x-intercept is fundamental in understanding the behavior of rational functions. The x-intercept is the point where the graph of the function crosses the x-axis. At this point, the value of the function is zero, meaning that the y-value is zero. To find the x-intercept of a rational function like \( f(x) = \frac{4x+3}{x-7} \), you need to focus on the numerator of the function.
Here's what you do:

• Set the numerator equal to zero: \( 4x + 3 = 0 \).
• Solve for \( x \), which gives you \( x = -\frac{3}{4} \).

Thus, the x-intercept is \((-\frac{3}{4}, 0)\). This means when \( x = -\frac{3}{4} \), the function's output is zero, indicating it crosses the x-axis at this point. Knowing how to determine the x-intercept is crucial for sketching or analyzing the graph, as it provides insight into where the function might change signs or contribute to its overall shape.

Y-intercept

The y-intercept of a function is where the graph intersects the y-axis. In other words, it's the point at which the function has an x-value of zero. For rational functions, identifying the y-intercept involves substituting \( x = 0 \) into the function equation.
Consider the example \( f(x) = \frac{4x+3}{x-7} \). Here's how to find the y-intercept:

• Substitute \( x = 0 \) into the function: \( f(0) = \frac{4(0) + 3}{0 - 7} \)
• This simplifies to \( f(0) = \frac{3}{-7} = -\frac{3}{7} \).

Therefore, the y-intercept is \((0, -\frac{3}{7})\). At this point, the function intersects the y-axis. The y-intercept plays a significant role in giving a sense of where the function starts graphing, especially when plotting the graph. It also shows how the function behaves when approaching zero from either direction on the x-axis.

Asymptote

Asymptotes are lines that the graph of a function approaches but never actually touches or crosses. In the context of rational functions, you may encounter both vertical and horizontal asymptotes.Firstly, let's explore **vertical asymptotes**:

• Vertical asymptotes occur where the function's denominator equals zero, as the function becomes undefined at these points.
• For \( f(x) = \frac{4x+3}{x-7} \), set the denominator \( x - 7 = 0 \) and solve for \( x \).
• This gives \( x = 7 \), meaning there is a vertical asymptote at \( x = 7 \).

Next, let's consider **horizontal asymptotes**:

• To find horizontal asymptotes, compare the degrees of the numerator and denominator polynomials.
• In the case of \( f(x) = \frac{4x+3}{x-7} \), both have degree 1.
• Therefore, the horizontal asymptote is calculated as \( y = \frac{4}{1} = 4 \).

Horizontal and vertical asymptotes are crucial for understanding the end-behavior of rational functions. They guide how the function behaves as \( x \) approaches certain critical values or extends towards infinity.