analyze the graph of the following function.

r(x)= x^2+x-6
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x^2-x-2

there are 8 steps but they don't make sense to me... please show work!! thanks

what are the 8 steps?

I assume they involve finding asymptotes, roots, intercepts, etc.

To get started,
x^2+x-6 = (x+3)(x-2)
x^2-x-2 = (x+1)(x-2)

To analyze the graph of the function, we can follow these steps:

1. Factorize the numerator and denominator of the rational function. This helps us identify any potential holes, vertical asymptotes, or horizontal asymptotes.
r(x) = (x + 3)(x - 2) / (x + 1)(x - 2)

2. Identify any vertical asymptotes. These occur at the values of x for which the denominator of the rational function equals zero. In this case, there is a vertical asymptote at x = -1.

3. Determine any holes in the graph. Holes occur when there is a common factor between the numerator and denominator that can be canceled out. Here, we have a (x - 2) term in both the numerator and denominator. Thus, there is a hole at x = 2.

4. Find the x-intercepts by setting the numerator equal to zero and solving for x. Here, we have x + 3 = 0, which gives us x = -3.

5. Determine the y-intercept by evaluating r(x) when x = 0. Plugging in x = 0 into the function, we get r(0) = (-3)(-2) / (1)(-2) = 3.

6. Identify any horizontal asymptotes. To determine the horizontal asymptotes, we examine the degrees of the numerator and denominator. Since both have the same degree (2), we look at the ratio of the leading coefficients. In this case, the leading coefficients are both 1. Therefore, there is a horizontal asymptote at y = 1.

7. Draw the graph, taking into account all the information gathered from the previous steps, such as vertical asymptotes, holes, x-intercepts, y-intercept, and horizontal asymptotes.

8. Label the graph with the corresponding points obtained in previous steps.

By following these steps, you will be able to analyze the graph of the given function r(x).