SOLUTION: Write the equation of a parabola with x-intercepts at (3, 0) and (9, 0) that passes through the point (10,-7)

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Question 1127572: Write the equation of a parabola with x-intercepts at (3, 0) and (9, 0) that passes through the point (10,-7)
Found 3 solutions by josmiceli, greenestamps, josgarithmetic:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
The general formula is:
+y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+
( 3,0 )
( 9,0 )
( 10,-7 )
------------------------------
( 3,0 )
+0+=+a%2A3%5E2+%2B+b%2A3+%2B+c+
(1) +9a+%2B+3b+%2B+c+=+0+
-------------------------------
(9,0)
+0+=+a%2A9%5E2+%2B+b%2A9+%2B+c+
(2) +81a+%2B+9b+%2B+c+=+0+
-------------------------------
( 10,-7 )
+-7+=+a%2A10%5E2+%2B+b%2A10+%2B+c+
(3) +100a+%2B+10b+%2B+c+%2B+7+=+0+
-------------------------------
Multiply both sides of (1) by +3+
and subtract (1) from (2)
+54a+-+3c+=+0+
+c+=+18a+
--------------------------
Plug this result into (3)
(3) +100a+%2B+10b+%2B+18a+%2B+7+=+0+
(3) +118a+=+-10b+-+7+
(3) +10b+=+-118a+-+7+
(3) +b+=+%281%2F10%29%2A%28+-118a+-+7+%29+
-------------------------------------
You can plug c and b back into (1) and
find a
Will let you finish



Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


Answer the question using the symmetry of the parabola and the vertex form of the equation of a parabola,

y-k+=+a%28x-h%29%5E2 where (h,k) is the vertex.

(1) With x-intercepts 3 and 9, the axis of symmetry is x=6, so the equation is of the form

y+=+a%28x-6%29%5E2%2Bk

(2) Use one of the x-intercepts and the third point in that equation to find the value of a:
    (9,0)   -->   0 = a(3^2)+k   9a+k =  0
    (10,-7) -->  -7 = a(4^2)+k  16a+k = -7
                             -----------
                                 7a   = -7
                                    a = -1


The equation is now of the form

y+=+-1%28x-6%29%5E2%2Bk

(3) Use either x-intercept to find k:

0+=+-1%289%29%2Bk
k+=+9

We are done....

ANSWER: y+=+-1%28x-6%29%5E2%2B9

Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
y=a%28x-3%29%28x-9%29

a=y%2F%28%28x-3%29%28x-9%29%29

The given point of (10,-7):
a=-7%2F%28%2810-3%29%2810-9%29%29
a=-7%2F%287%2A1%29
a=-1

-
y=-%28x-3%29%28x-9%29----------the equation in factored form