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Question 1167950: write the standard form of the equation of the parabola whose vertex is (-4,11) and passes through the point (-6,15).
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39623)   (Show Source):
You can put this solution on YOUR website!  because of the given vertex. Use the known given other point to help solve for "a". Do whatever else you need for the form of equation you want.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the standard equation of a parabola is ax^2 + bx + c = y
the equation goes through the point (-6,15)
replace x with -6 and y with 15 to get:
a * 36 + b * -6 + c = 15
simplify to get:
36a - 6b + c = 15
the equation also goes through the point (-4,11)
replace x with -4 and y with 11 to get:
a * 16 + b * -4 + c = 11
simplify to get:
16a - 4b + c = 11
you have two equations that need to be solved simultaneously.
they are:
36a - 6b + c = 15
16a - 4b + c = 11
subtract the second equation from the first to get:
20a - 2b = 4
the vertex of the parabola is at (-4,11)
the formula for the vertex in the standard form of the equation is:
x = -b/2a
when x = -4, this equation becomes;
-b/2a = -4
solve for b to get b = 8
the equation of 20a - 2b = 4 becomes:
20a - 16 = 4
add 16 to both sides to get:
20a = 20
solve for a to get:
a = 1
you have:
a = 1 and b = 8
go back to the original equations of:
36a - 6b + c = 15
16a - 4b + c = 11
in either equation, replace a with 1 and b with 8 to get:
36a - 6b + c = 15 becomes 36 - 48 + c = 15 which becomes -14 + c = 15.
solve for c to get c = 27.
you now have a = 1, b = 8 and c = 27
go back to your original equation of ax^2 + bx + c = y and replace a with 1 and b with 8 and c with 17 to get:
x^2 + 8x + 27 = y
graph this equation to get:
when you replace x with -4, the equation becomes:
(-4)^2 + 8*-4 + 27 = y
solve for y to get y = 11
when you replace x with -6, the equation becomes:
(-6)^2 + 8*-6 + 27 = y
solve for y to get y = 15
both points are on the parabola.
the vertex is at x = -b/2a which becomes x = -8/2 which becomes x = -4
the y value was already calculated and is equal to 11 when x = -4.
you can graph the equation to get:
both points are on the parabola and the vertex is at (-4,11).
solution is confirmed to be good.
the standard form of the equaion of the parabols is:
y = x^2 + 3x + 27
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