Question

What is the equation of the parabola that has a vertex at `(7, 3)` and passes through point `(3, -2)`?

Answer 1

`f(x) = -(5x^2)/33 + (70x)/33 - 3`

Explanation:

Standard form of a quadratic function
`f(x) = ax^2 + bx + c`
There are 3 unknowns a, b and c. We need 3 equations.
a. x-coordinate of vertex = 7
`x = - b/(2a)` --> `7 = - b/(2a)` --> `b = - 14a`(1)
b. y-coordinate of vertex = 3
y vertex = f(7) = 3 --> 3 = 49a + 7b + c (2)
c. The parabola passes at point (3, -2)
f(3) = -2 = 9a + 3b + c (3)
Now, solve the system of 3 equations (1), (2) and (3) to get a, b, and c
Equation (2) --> 3 = 49a + 7b + c = 49a - 98a + c = - 49a + c (2)'
Equation (3) --> - 2 = 9a + 3b + c = 9a - 42a + c = - 33a + c (3)'
Subtract (3)' from (2)', we get: - 33a = 5 -->
`a = - 5/33`.
`b = 70/33`
c = -3
`f(x) = - (5x^2)/33 + (70x)/33 - 3`