Follow the steps for graphing a rational function to graph the function .

Question

Follow the steps for graphing a rational function to graph the function

R(x)=3x+35x+20.

Use the real zeros of the numerator and denominator of R to divide the​ x-axis into intervals. Determine where the graph of R is above or below the​ x-axis by choosing a number in each interval and evaluating R there. Select the correct choice below and fill in the answer​ box(es) to complete your choice.

A.

The graph of R is above the​ x-axis on the​ interval(s)

enter your response here.

​(Type your answer in interval notation. Use a comma to separate answers as​ needed.)

B.

The graph of R is above the​ x-axis on the​ interval(s)

enter your response here

and below the​ x-axis on the​ interval(s)

enter your response here.

​(Type your answers in interval notation. Use a comma to separate answers as​ needed.)

C.

The graph of R is below the​ x-axis on the​ interval(s)

enter your response here.

​(Type your answer in interval notation. Use a comma to separate answers as​ needed.)

Answer 1

Here is an example of how to graph the rational function f(x) = (x^2 - 4) / (x - 2):

1. The domain is all real numbers except x = 2, which is the vertical asymptote.
2. To find the x-intercepts, we set the numerator equal to zero and solve: x^2 - 4 = 0 (x + 2)(x - 2) = 0 x = -2 or x = 2 So there are x-intercepts at (-2, 0) and (2, 0).
3. To find the y-intercept, we set x = 0 and evaluate the function: f(0) = (0^2 - 4) / (0 - 2) = 2 So the y-intercept is (0, 2).
4. As x approaches positive or negative infinity, the function approaches the line y = x, because the degree of the numerator is equal to the degree of the denominator.
5. There is a horizontal asymptote at y = x, because the degrees of the numerator and denominator are equal.
6. We plot the vertical asymptote x = 2, the x-intercepts (-2, 0) and (2, 0), the y-intercept (0, 2), and the horizontal asymptote y = x.
7. As x approaches 2 from the left, the function becomes very negative, and as x approaches 2 from the right, the function becomes very positive.
8. We sketch the graph by connecting the points and curves together, and keeping in mind the behavior of the function at the vertical and horizontal asymptotes. The graph looks like this.