Answer
See the explanation
Work Step by Step
Sure, here are the steps to graph a rational function:
1. Identify the Function: Determine the rational function you want to graph. A rational function is a function that can be expressed as the quotient of two polynomial functions.
2. Find Asymptotes: Determine the vertical asymptotes by finding the values of \( x \) for which the denominator of the rational function equals zero. Horizontal asymptotes can be found by comparing the degrees of the numerator and denominator polynomials.
3. Find Intercepts: Locate the x-intercepts by setting the numerator equal to zero and solving for \( x \). Find the y-intercept by evaluating the function at \( x = 0 \).
4. Analyze Behavior: Determine the behavior of the function as \( x \) approaches positive or negative infinity. This helps in sketching the graph accurately.
5. Plot Points: Choose additional points to plot on the graph, especially around the intercepts and asymptotes, to understand the behavior of the function between these points.
6. Sketch the Graph: Using all the information gathered from the previous steps, sketch the graph of the rational function.
Let's illustrate these steps with an example:
Consider the rational function \( f(x) = \frac{2x^2 - 3x - 2}{x - 2} \).
1. Identify the Function: We have the rational function \( f(x) = \frac{2x^2 - 3x - 2}{x - 2} \).
2. Find Asymptotes: The vertical asymptote occurs at \( x = 2 \) since the denominator equals zero at that point. There is no horizontal asymptote because the degree of the numerator (2) is equal to the degree of the denominator (1).
3. Find Intercepts: To find the x-intercepts, we solve \( 2x^2 - 3x - 2 = 0 \), which yields \( x = -1 \) and \( x = 2 \) (since \( x = 2 \) is also a root of the numerator, it cancels out). The y-intercept is found by evaluating \( f(0) = \frac{-2}{-2} = 1 \).
4. Analyze Behavior: As \( x \) approaches positive or negative infinity, the function will approach the line \( y = 2x \) (using the leading terms of the numerator and denominator).
5. Plot Points: Choose additional points to plot on the graph to understand its behavior. For example, you could pick \( x = 1 \) and \( x = 3 \).
6. Sketch the Graph: Using the information gathered, sketch the graph of the rational function, showing the asymptotes, intercepts, and behavior.
