Problem 89 - Algebra and Trigonometry

Question

Graph: $$f(x)=\frac{4 x-4}{x-2}$$ (Section 3.5, \text { Example } 5)

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{4 x-4}{x-2}\) consists of a horizontal line drawn at \(y=4\) with a hole at \(x=2\). The graph has no x-intercept, and the y-intercept is at \(y=4\). It has a vertical asymptote at \(x=2\).

Step 1: Simplify the function

We initially notice that we can factor out 4 out of the numerator of the function, giving: \(f(x)=\frac{4 (x-1)}{x-2}\). We can then simplify the function to \(f(x)=4\) when \(x≠2\).

Step 2: Determine important features of the graph

There are three major features we need to consider: The y-intercept, the x-intercept, and any asymptotes. The y-intercept of a function is the value of the function when \(x = 0\), which in this case is \(f(0)= 4\). The x-intercept is the value/s of \(x\) where the function equals 0, but for this function there are no such values. The function has an asymptote at \(x=2\), because this is where the function is undefined.

Step 3: Draw the graph

The final step is to draw the graph with the function \(f(x)=4\), for all \(x≠2\), a gap or hole at \(x=2\), and asymptote at \(x=2\). We will start by drawing a straight horizontal line at \(y=4\) (because \(f(x)=4\) for all \(x≠2\)), then introduce a gap or hole at \(x=2\) to represent the fact the function is undefined there, and then sketch a vertical asymptote at \(x=2\).

Key Concepts

Asymptotes

Understanding asymptotes is essential when graphing rational functions. An asymptote is essentially a line that the graph of a function approaches but never actually reaches. There are two main types of asymptotes: vertical and horizontal.

In the given exercise, the function \( f(x)=\frac{4x-4}{x-2} \) has a vertical asymptote. A vertical asymptote occurs at values of \( x \) where the function is undefined due to a zero denominator. In our case, when \( x=2 \) the denominator becomes zero, hence, the function is undefined, marking \( x=2 \) as the vertical asymptote for this graph.

Identifying Vertical Asymptotes

While graphing, this means that as \( x \) approaches 2, the values of \( f(x) \) grow without bound. It's important to note that the graph does not touch or cross this line. Graphically, we represent this by a dashed vertical line at \( x=2 \) and do not draw the graph through it. Remember, the presence of a vertical asymptote also indicates that there will be a restriction in the domain of the function.

Y-Intercept

The y-intercept is a fundamental point where the graph crosses the y-axis. The y-intercept is found by evaluating the function when \( x = 0 \). For our function \( f(x)=\frac{4x-4}{x-2} \) if we set \( x \) to 0, we will find the y-intercept to be \( f(0) = \frac{-4}{-2} = 2 \), thus the y-intercept is \( (0,2) \).

Plotting the Y-Intercept

On the Cartesian plane, this point is particularly easy to find and plot because you simply rise or fall to the correct value on the y-axis. When plotting or sketching graphs, starting with the y-intercept provides you with a reference point. Since rational functions can have transformations that shift the graph up, down, left, or right, knowing the y-intercept helps us understand these shifts.

X-Intercept

The x-intercept is where the graph crosses the x-axis. These intercepts are found by determining the values of \( x \) that make the function equal to zero. For rational functions like \( f(x)=\frac{4x-4}{x-2} \), this entails setting the numerator equal to zero and solving for \( x \).

However, the given function simplifies to \( f(x)=4 \) for all \( x \) not equal to 2, which means that there is no value of \( x \) for which the function equals zero. Hence, this function does not have an x-intercept.

Importance of X-Intercepts

When a graph does have x-intercepts, they are crucial for understanding the behavior of the function since they indicate where the function's output changes sign. In this case, since there are no x-intercepts, we understand that the function does not cross or touch the x-axis at any point.