Zand G. answered • 08/20/23
Patient SAT, ACT, and Math Tutor from Carnegie Mellon University
Hi Gabe, to answer this problem you would first want to simplify the function.
Factor out common terms from the numerator and the denominator: x*(x + 4) / 3x*(-x^2 + x + 6). Notice here that I just factored out an x on the numerator and a 3x on the denominator.
1.) To identify the holes, you need to find a common factor that can be canceled out in both the numerator and denominator. The factor x is common (notice how there is an x in both x and 3x). As x approaches 0, the function becomes undefined (notice how plugging in x = 0 into the simplified function yields 0/0). So, there's a hole at x = 0.
2.) To identify the intercepts, you need to find both x- and y-intercepts. For the x-intercepts, these occur when f(x) = 0, which means the numerator equals 0. If you recall, our f(x) is our simplified function
x*(x + 4) / 3x*(-x^2 + x + 6). Setting x*(x + 4) = 0, this gives x =0 and x = -4 for the x-intercepts.
For the y-intercepts, these occur when x = 0. Plugging x = 0 into our simplified function, we get f(0) =
0*(0 + 4) / 3*0*(-0^2 + 0 + 6), which is undefined. Thus, there are no y-intercepts.
3.) To identify the horizontal asymptote, you want to compare the degrees of the highest powers of x in the numerator and the denominator. You want to see what happens to f(x) as x becomes really large. Notice how the highest power of x on the denominator is x^3 and the highest power of the numerator is x^2. As x becomes really large, the denominator will grow much faster than the numerator, so the horizontal asymptote is at y = 0 (as x approaches infinity, x^2/x^3 will approach 0).
To graph the function, keep in mind the key features you calculated for the function. The function has a hole at x = 0. There's an x-intercept at (-4,0). The horizontal asymptote is at y = 0. You can check your sketch against a graphing calculator. Feel free to message me with any questions about your sketch. Hope that helped!
