Question 178079
We know that the line of symmetry is x=-2 and that one intercept is at (7,0) which is 9 units from the line of symmetry which passes thru (-2,0). if we travel 9 units in the opposite direction then the other x intercept would fall on the point (-11,0)
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If you use the formula for a parabola to figure out its equation you would use,
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{{{(y-k)=a(x-h)^2}}}where (h,k) is the vertex and a is the vertical stretch
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we have the vertex and a point on that vertex ..therefore we can find a
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{{{y+4=a(x+2)^2}}}
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using (7,0) as our known point
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{{{0+4)=a(7+2)^2}}}
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{{{4=81a}}}
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a=4/81
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therefore our equation for this parabola is :{{{ y+4=(4/81)(x+2)^2}}}
or written with y isolated y=(4/81)(x+2)^2-4}}}
and graphing we see that indeed the x intercepst appear to be where we have them.
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The reason we cant use just the roots to find the equation of a  parabola is as follows:

The parabola (a quadratic) that forms from these roots is the product of these binomials... and a CONSTANT MULTIPLIER which is sometimes called the VERTICAL STRETCH OR COMPRESSION. Many different shapes of parabolas can have the same roots. Heres our case

{{{c(x+11)(x-7)=c(x^2+4x-77)=cx^2+4cx-77c}}}
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The 'c' doesn't effect the roots... but it does effect the vertical compression or stretch of the parabola.

We cannot write a more specific parabola equation without knowing more information... like vertex, focus, directrix, or another point. We had our extra point and using the parabola formula found our vertical stretch.
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If we were only given the roots,that is two pieces of information (two points)... but a parabola is a curve. There is no way to extrapolate exact shape based on two points alone (you can only find the equation of a line with two points)



{{{graph(400,400,-15,15,-15,15,(4/81)(x+2)^2-4)}}}