Problem 36 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{8}{4-x}$$

Step-by-Step Solution

Answer

The graph of function \(f(x) = \frac{8}{4-x}\) has a vertical asymptote at x = 4 and a horizontal asymptote at y = 0. The y-intercept of the function is at y = 2 and there is no x-intercept.

Step 1: Define the Function and Find the Asymptotes

Given function is \(f(x)=\frac{8}{4-x}\). The denominator is set to zero and solved to find vertical asymptote(s). Therefore, the vertical asymptote is at x = 4. The horizontal asymptote of any rational function in which the degree of the numerator is less than the degree of the denominator, is always the x-axis i.e, y = 0.

Step 2: Find the X and Y Intercepts

The x-intercept is the value of x for which f(x) = 0. However, for this function, there is no existence of x-intercept as the numerator of the rational function, 8, cannot be zero. The y-intercept is the value of y when x = 0. Therefore, putting x = 0 in function, we get f(0) = 2, so the y-intercept is at y = 2.

Step 3: Sketch the Graph

Firstly, plot the asymptotes on the xy-plane at x = 4 (vertical line) and y = 0 (horizontal line). Since there are no x-intercepts, no point is marked on the x-axis. However, put a point at y = 2 on the y-axis for the y-intercept. The curve of the function will be approaching but not touching the asymptotes. As x approaches 4 from the left, y approaches negative infinity and as x approaches 4 from the right, y approaches positive infinity. As x goes to positive or negative infinity, y will approach the x-axis but never cross it, this indicates that the horizontal asymptote is indeed y = 0.

Key Concepts

Asymptotes

An asymptote is a line that a graph approaches but never actually touches. In rational functions like the one we are working with, asymptotes play a crucial role. They can be vertical or horizontal, offering valuable insights into the behavior of the graph. To find a **vertical asymptote** in a rational function, you look at where the denominator equals zero, as the function becomes undefined. For the function \( f(x)=\frac{8}{4-x} \), setting \( 4-x = 0 \) gives us \( x = 4 \) as a vertical asymptote. This means as \( x \) approaches 4, the graph will shoot up to infinity or down to negative infinity.

Horizontal asymptotes help us understand the behavior of the function as \( x \) goes to very large positive or negative values. To determine a **horizontal asymptote**, compare the degrees of the polynomial in the numerator and the denominator. In this case, since the degree of the numerator (0) is less than the degree of the denominator (1), the horizontal asymptote is \( y = 0 \). This indicates that as \( x \) becomes very large or very small, the value of \( f(x) \) will get closer and closer to 0, but never actually reach it.

X and Y Intercepts

Intercepts are the points where the graph intersects the axes. Knowing where these intersections occur can help you plot key points on the graph. For rational functions, finding these points is quite straightforward.

**X-intercepts** occur where the graph crosses the x-axis, meaning \( f(x) = 0 \). However, for our function \( f(x)=\frac{8}{4-x} \), the numerator is constant and non-zero, hence the function never crosses the x-axis. Thus, there are no x-intercepts.

On the other hand, **Y-intercepts** are found by setting \( x = 0 \) and solving for \( y \). Substituting \( x = 0 \) in \( f(x)=\frac{8}{4-x} \) gives \( f(0) = 2 \). Therefore, the y-intercept is at the point (0, 2). This is where the graph will cross the y-axis.

Graph Sketching

Sketching the graph of a rational function involves a few key steps: plot the intercepts, identify and draw the asymptotes, and then visualize the curve's behavior as it approaches these lines. For \( f(x)=\frac{8}{4-x} \), follow these steps:

• Start by drawing the vertical asymptote as a dashed line at \( x = 4 \). This will guide the steep behavior of the graph as \( x \) nears this value.
• Next, draw the horizontal asymptote at \( y = 0 \). Remember, the curve will approach but never touch this line.
• Plot the y-intercept at (0, 2) on the graph. This is a key point that will help shape the rest of the curve.
• The graph will approach negative infinity as \( x \) approaches 4 from the left, and positive infinity from the right, hugging closely to the vertical asymptote.
• As \( x \) moves far from 4, in both directions to positive or negative infinity, the curve will get closer to the horizontal asymptote at \( y = 0 \), flattening out but never intersecting the x-axis.

Visualizing these behaviors helps create an accurate sketch of the graph, illustrating how the rational function behaves across its domain.