Identifying the formula for a quartic graphic

Question

I am attempting to help someone with their homework and these concepts are a bit above me. I apologize for the terrible graph drawing. I am using a surface pro 3 and it has an awful camera so I can't take a picture of the problem so I attempted to trace it.

https://i.stack.imgur.com/FzMSf.png

The problem shows a graph in the shape of a W. The left part comes downward to -3,0, the center is at 0,0 and the right is at 3,0. There are no numbers shown on the y axis. I believe the W shape indicates that it is a graph of a quartic equation.

The problem states:

"Find the formula for the graph above, given that it is a polynomial, that all zeroes of the polynomial are shown, that the exponents of each of the zeroes are the least possible, and that it passes through the point (-1, -8)."

Now from what I could find trying to research quartic equations is that my formula should look like ax^4+bx^3+cx^2+dx+e, but I have no idea where to start.

Edit: I managed to get the solution as x^4-9x^2 by using Desmos and playing with the graph using your comments to guide me, but I am not sure how to go through the steps mathematically.

Answer 1

The graph shows roots at $x = -3, 0, 3$, and there's a double root at zero due to the way the graph is tangent to the y-axis at that point. The equation is therefore the product of the four factors $(x + 3)(x - 3)(x + 0)(x + 0) = (x^2 - 9)(x)(x) = x^4 - 9x^2$, possibly with another constant scaling multiplier. But if we check $x = -1$ for this expression, then we see $x^4 - 9x^2 = (-1)^4 - 9(-1)^2 = 1 - 9 = -8$, which meets the final constraint, and so no constant multiplier is needed.

Answer 2

If this is a quartic then you're given the three roots of its derivative. You can integrate this cubic to recover the quartic and use the known point to find the leading coefficient and the constant of integration. Note that $x=0$ is a zero of both the quartic and the cubic.

Solution:

The derivative has roots $-3,0,3$ and so is $ax(x^2-9)$. The quartic is thus $a (x^4/4-9 x^2/2)+ c$. Since $0$ is a root, $c=0$. Now solve $-8=a(1/4-9/2)$ to find $a$.