Answer:
1) Place a point on (-4, 0) and (2, 0).
2) Draw a vertical line at x = -1 and x = 3.
3) Draw a horizontal line at y = 1.
Explanation:
We are given the factored form of a rational function and asked to graph the x-intercepts, the vertical asymptotes, and the horizontal asymptote.
1) X-Intercepts
Since this function is given to us in factored form, we can easily find the x-intercepts. These are points at which the numerator equals 0.
0 = (x - 2)(x + 4)
x = -4, 2
Place a point on (-4, 0) and (2, 0).
2) Vertical Asymptotes
Again, we can easily find the vertical asymptotes because the function is given in factored form. These are the places at which the denominator equals 0, because dividing by zero results in an undefined value.
0 = (x - 3)(x + 1)
x = -1, 3
Draw a vertical line at x = -1 and x = 3.
3) Horizontal Asymptote
Lastly, we will find the horizontal asymptote. For this, we will look at the non-factored form of the function. There are three different possibilities.
1 (n < d) - If the numerator's degree is less than the denominator's degree, then the horizontal asymptote is y = 0.
2 (n = d) - If the numerator's degree and denominator's degree are the same, then the horizontal asymptote is a ratio of the leading coefficients of the numerator and the denominator.
3 (n > d) - If the numerator's degree is greater than the denominator's degree, then there is no horizontal asymptote.
Here, the numerator's degree and denominator's degree are the same, we so it is the ratio of the leading coefficients.
y =
➜ y = 1
Draw a horizontal line at y = 1.
4) Full Graph
I have graphed this function so you can see all of the pieces we just found. See attached.
You can see that the function crosses at the x-intercepts, that it never crosses the vertical asymptotes, and that the horizontal asymptote affects the shape and end behavior.