Problem 34 - A Graphical Approach to College Algebra

Question

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{6-3 x}{4-x}$$

Step-by-Step Solution

Answer

Vertical asymptote at \(x=4\), horizontal asymptote at \(y=3\), intercepts at \((2,0)\) and \((0, \frac{3}{2})\).

Step 1: Understanding the Function

The function is given by \( f(x) = \frac{6-3x}{4-x} \). This is a rational function, which means it is the ratio of two polynomials.

Step 2: Finding Vertical Asymptotes

A vertical asymptote occurs when the denominator of the function is zero and the numerator is not zero at that point. Set \(4-x=0\) to find the vertical asymptote: \[4-x=0 \rightarrow x=4.\] So, there is a vertical asymptote at \(x=4\).

Step 3: Finding Horizontal Asymptotes

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Both polynomials have degree 1. The horizontal asymptote is the ratio of the leading coefficients: \(-\frac{3}{-1} = 3\). Thus, the horizontal asymptote is \(y=3\).

Step 4: Finding the x-intercept

The x-intercept is found by setting the numerator equal to zero and solving for \(x\). So, \(6-3x = 0\): \[6-3x = 0 \rightarrow 3x = 6 \rightarrow x = 2.\] Thus, the x-intercept is at \((2,0)\).

Step 5: Finding the y-intercept

The y-intercept is found by evaluating the function at \(x=0\). \[f(0) = \frac{6-3\times0}{4-0} = \frac{6}{4} = \frac{3}{2}.\] Therefore, the y-intercept is at \((0, \frac{3}{2})\).

Step 6: Sketching the Graph

To sketch the graph, plot the intercepts \((2,0)\) and \((0, \frac{3}{2})\), and draw the vertical asymptote at \(x=4\) and the horizontal asymptote at \(y=3\). The function will approach these asymptotes but never touch or cross them at infinity.

Key Concepts

Vertical Asymptotes

In the realm of rational functions, vertical asymptotes represent values where the function "blows up"—essentially where it becomes undefined. They occur when the denominator equals zero but the numerator does not. For the function \(f(x) = \frac{6-3x}{4-x}\), the denominator \(4-x\) becomes zero at \(x=4\). This means we have a vertical asymptote at \(x=4\). At this line, the graph will never cross. Instead, as \(x\) approaches 4 from both the left and right, the function values increase or decrease toward infinity. Vertical asymptotes are crucial for understanding the behavior of a graph in rational functions, giving you insight into where rapid changes occur.

Horizontal Asymptotes

Horizontal asymptotes indicate the value that the function approaches as \(x\) moves toward infinity or negative infinity. They're particularly useful for visualizing the behavior of rational functions over large distances. To find the horizontal asymptote of \(f(x) = \frac{6-3x}{4-x}\), compare the degrees of the polynomials in the numerator and denominator. Here, both have a degree of 1. When the degrees are the same, the horizontal asymptote is found using the ratio of the leading coefficients—in this case, \(-3\) from the numerator and \(-1\) from the denominator, yielding \(y = 3\). This means that as \(x\) heads towards either extreme, the values of \(f(x)\) tend toward 3, but do not actually become equal to 3, especially at infinity. In graphing, the horizontal asymptote is represented by a line that the curve approaches but never touches.

Intercepts

Intercepts are the values where the graph crosses the axes and are key points in sketching graphs. There are two types to consider: x-intercepts and y-intercepts.

• **X-Intercepts** occur where the function equals zero. For \(f(x) = \frac{6-3x}{4-x}\), setting \(6-3x = 0\) gives \(x = 2\). Hence, the x-intercept is at the point \((2,0)\), where the graph intersects the x-axis.
• **Y-Intercepts** occur where the graph crosses the y-axis. This happens when \(x = 0\). Plugging into the function yields \(f(0) = \frac{6}{4} = \frac{3}{2}\), giving the y-intercept at \((0, \frac{3}{2})\).

Mapping these intercepts on a graph helps to provide key anchor points for drawing the curve. Intercepts are invaluable for a clearer and more structured understanding of rational functions.