Given, the vertex of a parabola is (2,7).
We know the the general equation of vertex form of parabola is
[tex] y = a(x-h)^2+k [/tex], where (h,k) is the vertex of the parabola.
Here the value of h = 2, k = 7. So by substituting the values of h and k in the general equation we will get,
[tex] y = a(x-2)^2+7 [/tex]
Given the parabola passes through the point (-1,3). So we will substitute x = -1 and y = 3 in this equation to find a. We will get,
[tex] 3 = a(-1-2)^2+7 [/tex]
[tex] 3 = a(-3)^2+7 [/tex]
[tex] 3 = a(9) +7 [/tex]
[tex] 3 = 9a+7 [/tex]
Now we can get a by moving 7 to the left side by subtracting it from both sides. We will get,
[tex] 3-7 = 9a+7-7 [/tex]
[tex] -4 = 9a [/tex]
[tex] 9a = -4 [/tex]
Now to get a we will move 9 to the right side by dividing it to both sides. We will get,
[tex] \frac{9a}{9} =\frac{(-4)}{9} [/tex]
[tex] a = -\frac{4}{9} [/tex]
So the equation of the parabola is,
[tex] y =-\frac{4}{9} (x-2)^2+7 [/tex]
To simplify the equation we will multiply both sides by 9. We will get,
[tex] 9y = (9)(-\frac{4}{9}(x-2)^2+7)) [/tex]
When we distribute 9 to the right side we will get,
[tex] 9y = -4(x-2)^2+63 [/tex]
[tex] 9y = -4(x^2-4x+4)+63 [/tex]
[tex] 9y = -4x^2+16x-16+63 [/tex]
[tex] 9y = -4x^2+16x+47 [/tex]
So we have got the required equation of parabola.
