Question

You should now have three zeros as well as two points between those zeros for a total of 5 points for your graph. You are now going to use those 5 points to create a graph of your function. Use the link below to access a Desmos template where you will enter those 5 points into a table.

what do I do for -13x^2(5x+10)(x-3) to find the other points besides the zeros I need 5 points

Answer 1

First identify the zeros (x-values where the expression equals 0):

- From x^2 you get x = 0 (multiplicity 2)
- From 5x + 10 = 0 you get x = −2
- From x − 3 = 0 you get x = 3

So the three x-intercepts are (−2, 0), (0, 0), and (3, 0). Now pick one x between −2 and 0, and one x between 0 and 3, and evaluate the function there. The function simplifies to
f(x) = −13 x^2 (5x + 10)(x − 3) = −65 x^2 (x + 2)(x − 3).

Evaluate at x = −1 (between −2 and 0):
f(−1) = −65(−1)^2(−1 + 2)(−1 − 3) = −65·1·1·(−4) = 260
Point: (−1, 260)

Evaluate at x = 1 (between 0 and 3):
f(1) = −65(1)^2(1 + 2)(1 − 3) = −65·1·3·(−2) = 390
Point: (1, 390)

So the five points you can enter into Desmos are:
- (−2, 0)
- (−1, 260)
- (0, 0)
- (1, 390)
- (3, 0)

Enter these x and y values into the Desmos table. (Note: since x = 0 has multiplicity 2 the graph will “bounce” at the x-axis there. Also the leading term is −65x^4, so both ends of the graph go down.)