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How to find A value in factored form?

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Two points define a line, but you need three points to define a parabola. If you only have the one x intercept and the y intercept, then there will be infinite parabolas that pass through them. If you have both x intercepts, then your should be okay.

In the form y=a(x-r)(x-s), the two x intercepts (aka, zeroes) are r and s. Now, let’s multiply it out to get:

y = ax^2 - a(r+s)x + ars

Let’s say the y intercept is K. That is, the parabola passes through the point (0,K) which means that when x=0, y=K. So, plug in those two values for x and y to get:

K = a(0)^2 - a(r+s)(0) + ars

K = ars

a = K/(rs)

Remember, K is the y intercept and r and s are the two x intercepts (zeroes). If you have all three, the a=K/(rs). But if you only know one of the zeroes, then there are infinitely many possible values for a, because there are infinitely many possible values for the other zero.

One exception worth noting is if one of the zeroes is 0, because the formula above won’t work. In that case, the parabola goes through (0,0) and thus the y intercept is also 0.

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