SOLUTION: Hi can someone help me with determining the equation of the quadratic function with a vertex of (3,5) and point (5,1) and a parabola with the points (-1,4), (1, -2) and (2,1)

Question 1138: Hi can someone help me with determining the equation of the quadratic function
with a vertex of (3,5) and point (5,1)
and a parabola with the points (-1,4), (1, -2) and (2,1)
Thanks!
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
Since the quadratic equation with vertex (a,b) should be as:
y - b = c (x-a)^2,
Now (a,b) = (3,5) and (5,1)is on the graph, we have
1 - 5 = c (5 -3)^2 = 4 c
and so c = -1, the quadratic equation as
y - 5 = - (x - 3)^2
Next, let the equation of the parabola be y = a x^2 + bx + c
Since it passing through (-1,4), (1, -2) and (2,1).
We have
a - b + c = 4 ...(1)
a + b + c = -2 ...(2)
4a + 2b + c = 1 ...(3)
By (2)-(1), we have 2b = -6, so b=-3.
Replace b by -3 in (1),we have a + c = 1...(4)
Replace b by -3 in (3),we have 4a + c = 7...(5)
By (5)-(4), we have 3a=6, a =2.

Replace a by 2 in (4),we have c= 1-2 = -1.
Hence,the equation of the parabola be y = 2 x^2 - 3x -1.

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