Find the eqeuation of the parabola that passes theough (3,0)(-5,0) and (5,-50)

Question

whpalmer4 · Answer

If the parabola passes through (3,0) and (-5,0), that means the equation of the parabola has roots at x = 3 and x = -5.  We can write the equation as
$$P(x) = k(x-3)(x+5)$$ where $k$ is a scaling factor to get the parabola to hit our target point at (5,-50).  Multiply out the polynomial, and find the value of $k$ such that $P(5) = -50$.

whpalmer4 · Answer

Actually, no need to multiply it out before finding $k$, probably easier if you don't!

anonymous · Answer

I'm a little confused how do you find k

whpalmer4 · Answer

$$P(5) = -50 = k(x-3)(x+5) = k(5-3)(5+5) = 20k$$$$k=?$$

anonymous · Answer

20?

whpalmer4 · Answer

Might be confusing to read with so many versions on the same line:
$$-50 = 20k$$
Solve for $k$

anonymous · Answer

k=-2.5

whpalmer4 · Answer

Exactly!  So what is your final equation?

whpalmer4 · Answer

(The value of k also controls whether the parabola opens up or down, of course)

anonymous · Answer

p(x)=-2.5(x-3)(x+5) ?

anonymous · Answer

Is that correct?

whpalmer4 · Answer

Yes, or (-5/2)(x^2+2x-15) or whatever.  Here's a plot of the final result:

whpalmer4 · Answer

If you squint just right you can see that the curve goes through (5,-50) and crosses the x-axis at x = -5 and x = 3.

anonymous · Answer

ok thanks

whpalmer4 · Answer

Does that make sense to you?