Problem 6 - 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) $$f(x)=\frac{4 x+3}{x-7}$$

Step-by-Step Solution

Answer

The horizontal asymptote is \(y=4\), matching option C.

Step 1: Understanding the Rational Function

The given function is \( f(x) = \frac{4x + 3}{x - 7} \). This is a rational function where the numerator is a linear polynomial \(4x + 3\) and the denominator is a linear polynomial \(x - 7\).

Step 2: Finding the x-intercept

To find the \(x\)-intercept of \(f(x)\), set the numerator equal to zero: \(4x + 3 = 0\). Solving for \(x\) gives \(x = -\frac{3}{4}\). Hence, the \(x\)-intercept is \((-\frac{3}{4}, 0)\), which does not match any options in Column II for an \(x\)-intercept.

Step 3: Finding the y-intercept

To find the \(y\)-intercept of \(f(x)\), evaluate \(f(0)\): \(f(0) = \frac{4(0) + 3}{0 - 7} = \frac{3}{-7} = -\frac{3}{7}\). Thus, the \(y\)-intercept is \((0, -\frac{3}{7})\), which does not match any options in Column II for a \(y\)-intercept.

Step 4: Identifying the Horizontal Asymptote

For rational functions where the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients. Here, both the numerator and denominator have a degree of 1, and the leading coefficients are 4 and 1 respectively. Therefore, the horizontal asymptote is \(y = \frac{4}{1} = 4\), matching option C.

Step 5: Identifying the Vertical Asymptote

The vertical asymptote occurs where the denominator is zero, provided it does not cancel with a factor in the numerator. Set \(x - 7 = 0\), giving \(x = 7\). Therefore, there is a vertical asymptote at \(x = 7\), which does not match any options in Column II for a vertical asymptote.

Step 6: Checking for Holes

A hole exists in the graph of a function if a factor in the denominator cancels with a factor in the numerator. Here, \(x - 7\) in the denominator does not cancel with any factor in the numerator \(4x+3\), hence there is no hole at \(x = -4\), and Option E does not apply.

Step 7: Checking for Oblique Asymptote

An oblique asymptote exists if the degree of the numerator is one more than the degree of the denominator. Here, both the numerator and denominator have the same degree, so there is no oblique asymptote. Option F does not apply.

Key Concepts

Horizontal Asymptotes

When exploring rational functions, the horizontal asymptote acts as a line that the function approaches but never quite reaches as you move far away from zero along the x-axis. This behavior usually occurs when a function has polynomial terms in both its numerator and denominator.
In general, horizontal asymptotes in rational functions can be determined by the degrees of these polynomials:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis (i.e., \(y = 0\)).
• If the degree of the numerator equals the degree of the denominator, the asymptote is the ratio of the leading coefficients of these polynomials.
• If the numerator's degree is greater than the denominator's, there won't be a horizontal asymptote; in some cases, there might be an oblique asymptote instead.

For the provided function \(f(x) = \frac{4x + 3}{x - 7}\):
Both the numerator and denominator are linear expressions, meaning their degrees are equal. This gives us a horizontal asymptote based on the leading coefficients:
The asymptote is \(y = \frac{4}{1} = 4\). This corresponds with Option C of the matching exercise.

Vertical Asymptotes

Vertical asymptotes occur where a rational function's denominator equals zero, making the function undefined at that particular x-value.
They act as lines that the graph will approach but never touch or cross.
Identifying vertical asymptotes involves solving for the values where the denominator is zero, provided these factors don't cancel with factors in the numerator.

• For \(f(x) = \frac{4x + 3}{x - 7}\), setting the denominator equal to zero gives \(x - 7 = 0\), resulting in \(x = 7\).
• This marks the location of the vertical asymptote, as there are no factors that cancel out.

This implies that as \(x\) approaches 7, the function heads towards positive or negative infinity, displaying a behavior typical of vertical asymptotes.
Unfortunately, this result does not align directly with any options in the given exercise.

Intercepts

Intercepts help us understand where a graph interacts with the axes. For rational functions, there are two main types of intercepts:

• x-intercepts: These are found by setting the numerator equal to zero and solving for \(x\), provided the result doesn't also make the denominator zero.
• y-intercepts: These occur where \(x = 0\). By substituting zero into the function, the y-intercept can be calculated: \(f(0) = \frac{4 \cdot 0 + 3}{0 - 7} = -\frac{3}{7}\).

In the context of our function, \(f(x) = \frac{4x + 3}{x - 7}\):
The x-intercept is found by solving \(4x + 3 = 0\), leading to \(x = -\frac{3}{4}\).
This produces an x-intercept at \((-\frac{3}{4}, 0)\).
The y-intercept calculation yields \((0, -\frac{3}{7})\).
Neither intercept exactly matches any given options in Column II for this exercise, indicating these features might not be the primary focus of the select correct descriptions for the function.