Problem 80 - Precalculus : Building Concepts and Connections

Question

Sketch a possible graph of a rational function \(r(x)\) of the following description: the graph of \(r\) has a horizontal asymptote \(y=-2\) and a vertical asymptote \(x-1,\) with \(y\) -intercept at (0,0).

Step-by-Step Solution

Answer

To construct the graph of the rational function \(r(x)\), begin by plotting the horizontal asymptote at \(y = -2\) and the vertical asymptote at \(x = 1\). Additionally, plot the y-intercept at (0,0). Given these guidelines, sketch the graph such that it approaches but does not cross the asymptotes.

Step 1: Draw the Asymptotes

The first step is to draw both the horizontal and vertical asymptotes. This gives us a framework in which the function will be graphed. So, draw a vertical line at \( x = 1 \), and a horizontal line at \( y = -2 \).

Step 2: Plot the y-intercept

The given function \( r(x) \) passes through the origin, i.e., (0,0). Mark this point on the graph.

Step 3: Sketch the graph

With the asymptotes as boundaries, sketch the graph. Ensure you have one portion of the graph in each of the two areas divided by the vertical asymptote. The graph should tend towards the horizontal asymptote but not cross it.

Key Concepts

Horizontal Asymptote

A horizontal asymptote in a rational function is a horizontal line that the graph of the function approaches as the value of the independent variable (usually x) goes to infinity or negative infinity. The equation given in the exercise specifies that there is a horizontal asymptote at \( y = -2 \). This means that as \( x \) becomes very large or very small, the graph of \( r(x) \) will get closer and closer to the line \( y = -2 \) but typically won't cross it.

• If the degree of the numerator of the rational function is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degrees are equal, the horizontal asymptote is the leading coefficient of the numerator divided by the leading coefficient of the denominator.
• If the numerator's degree is greater, there is no horizontal asymptote, but there may be an oblique asymptote.

Understanding horizontal asymptotes helps in predicting the long-term behavior of a function’s graph.

Vertical Asymptote

Vertical asymptotes occur in rational functions where the denominator is zero and the function tends to infinity. For the function described in the exercise, there is a vertical asymptote at \( x = 1 \). This means the function \( r(x) \) will tend either to positive or negative infinity as \( x \) approaches 1 from either the left or right side.

• To find the vertical asymptote, set the denominator equal to zero and solve for \( x \).
• Vertical asymptotes indicate a division by zero that the function cannot compute, making the graph of the function shoot up or down.
• The graph of the function will split around the vertical asymptote, forming separate branches on either side.

It's crucial to identify vertical asymptotes to avoid misrepresenting the behavior of the graph near undefined points.

Y-Intercept

The y-intercept of a graph is where the function crosses the y-axis, meaning \( x = 0 \). For the given rational function graph, this point is at \( (0, 0) \). The step-by-step solution indicates marking this point first on the graph.

• To find the y-intercept, substitute \( x = 0 \) into the function and solve for \( y \).
• This point provides a crucial reference, helping to anchor the sketch of the graph at one specific location.
• The y-intercept gives you an initial value and helps confirm the shape of the graph derived from both asymptotes.

Locating the y-intercept is an essential part of graphing functions because it establishes a known point on the graph before sketching the general shape.