First we need to factor. There is no way to simplify this function; no terms will cancel. We find no holes, but vertical asymptotes at *x* = -3 and *x* = -1: Since the degree of the numerator and degree of the denominator are the same, we have a horizontal asymptote at : The function is 0 when the numerator is 0, which occurs at *x* = -2 and *x* = 1: Now we need to figure out where the function is negative and where it is positive, which we can do with a numberline: : Finding *f*(0): We can add this point to the graph: We know some points of *f*, we know where it is 0, negative, and positive, and we know it must approach its asymptotes, so *f* must look like this: We haven't explained how we know that *f* has the shape it has, and we won't get to that until we start talking about concavity. For now, think about drawing the functions to make them "smooth," like MJ's criminal. Remember that intercepts are where a function crosses the axes. The *y*-intercept is *f*(0), if that exists. The *x*-intercepts are all values of *x* where the function is 0. |