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I know how to find the equation of a function like the following diagram that has y-intercept that is not the origin:

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But how I would find the equation of the function if instead of the curve cutting the y-axis at (0 -8), it cut the y axis at (0, 0)?

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If you add another root at x=0, but keep the left going to +inf and the right going to -inf, you need to pick one of the roots to be a double root (i.e. fifth-order polynomial). Depending on the shape of the graph, you'd know which one to make the double-root since it wouldn't cross the x-axis there.

For example, if you bring the -8 up to 0 to cross the x-axis, and the x=1 root "bounces" off the x-axis before going negative again at x=4, your equation would be something like -x*(x+1)*(x-1)^2*(x-4).

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