Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ q(x)=\frac{x-5}{3 x-1} $$

Step-by-Step Solution

Answer

Horizontal intercept: (5, 0); Vertical intercept: (0, 5); Vertical asymptote: x = 1/3; Horizontal asymptote: y = 1/3.

Step 1: Find the Horizontal Intercept

To find the horizontal intercept(s), set the numerator of the function equal to zero and solve for \(x\). The function is \(q(x) = \frac{x-5}{3x-1}\). The numerator is \(x-5\), set \(x - 5 = 0\). Solving this gives \(x = 5\). Thus, the horizontal intercept is at \((5, 0)\).

Step 2: Find the Vertical Intercept

To find the vertical intercept, evaluate the function at \(x = 0\). Substitute \(x = 0\) into the function: \(q(0) = \frac{0-5}{3 \cdot 0 - 1} = \frac{-5}{-1} = 5\). Thus, the vertical intercept is at \((0, 5)\).

Step 3: Find the Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero and the numerator is non-zero. Set the denominator \(3x - 1 = 0\) and solve for \(x\). \(3x = 1\) gives \(x = \frac{1}{3}\). Thus, there is a vertical asymptote at \(x = \frac{1}{3}\).

Step 4: Find the Horizontal Asymptote

For rational functions, if the degree of the numerator and the denominator is the same, the horizontal asymptote is the ratio of the leading coefficients. Here, the degrees are the same (both are 1). The leading coefficients are 1 for the numerator and 3 for the denominator, so the horizontal asymptote is \(y = \frac{1}{3}\).

Step 5: Sketch the Graph

With all critical points and asymptotes identified: plot the vertical asymptote at \(x = \frac{1}{3}\), the horizontal asymptote at \(y = \frac{1}{3}\), the horizontal intercept at \(x = 5\), and the vertical intercept at \(y = 5\). When sketching, ensure the curve approaches but never crosses the asymptotes, and it passes through the intercepts.

Key Concepts

horizontal intercept

The horizontal intercept of a rational function is found by determining where the numerator equals zero. In simple terms, this is where the output, or the value of the function, is zero. For the function \( q(x) = \frac{x-5}{3x-1} \), we focus on the numerator, which is \(x - 5\). By setting the numerator equal to zero, we solve the equation to find the horizontal intercept:

• Set \(x - 5 = 0\)
• Solve for \(x\): \(x = 5\)

This means the graph of the function crosses the x-axis at the point \((5, 0)\). It's the spot where the output value of the function is zero, implying the whole fraction equals zero at this point. Therefore, the horizontal intercept provides a vital clue about where the graph intersects the x-axis.

vertical intercept

The vertical intercept of a rational function is determined by evaluating the function when \( x = 0 \). It describes where the graph of the function crosses the y-axis. For our given function \( q(x) = \frac{x-5}{3x-1} \), substituting \( x = 0 \) yields the vertical intercept:

• Compute \(q(0) = \frac{0-5}{3 \cdot 0 - 1} = \frac{-5}{-1}\)
• The vertical intercept is at the point \((0, 5)\)

This coordinate \((0, 5)\) tells us where the graph intersects the y-axis. It's significant because it represents the output of the function when no x-value is inputted. Visually, this is where you would see the graph meet the y-axis, making it a crucial feature when sketching or interpreting the chart of the function.

vertical asymptote

Vertical asymptotes occur in rational functions when the denominator equals zero, and the numerator is not zero at that point. They represent values that make the function undefined, often signifying places the graph approaches but never actually touches. In \( q(x) = \frac{x-5}{3x-1} \), we find the vertical asymptote by setting the denominator to zero:

• Set \(3x - 1 = 0\)
• Solve for \(x\): \(x = \frac{1}{3}\)

So, there is a vertical asymptote at \(x = \frac{1}{3}\). Graphically, the curve gets very close to this line but does not cross or touch it, forming a seemingly invisible boundary. This feature reflects the idea that at \(x = \frac{1}{3}\), the function encounters a division by zero, which is undefined in mathematical terms.

horizontal asymptote

Horizontal asymptotes show the behavior of a function as \(x\) tends toward positive or negative infinity. They denote a level that the function’s graph approaches indefinitely. For the rational function \( q(x) = \frac{x-5}{3x-1} \), observe the degrees of the numerator and denominator, noting both are degree 1. This indicates that a horizontal asymptote exists:

• The leading coefficient of the numerator is 1.
• The leading coefficient of the denominator is 3.

Given that the degrees are identical, the horizontal asymptote is \(y = \frac{1}{3}\) (the ratio of the leading coefficients). This tells us that as \(x\) grows very large or very negative, the values of \(q(x)\) will get closer and closer to \(\frac{1}{3}\) but will not actually reach it. It's like a guideline that defines the long-term behavior of the curve in the function's graph.