# What is the equation of the parabola with a vertex at (2,3) and zeros at x=0 and x=4?

Dec 13, 2015

Find the equation of parabola

Ans: $y = - \frac{3 {x}^{2}}{4} + 3 x$

#### Explanation:

General equation: $y = a {x}^{2} + b x + c$. Find a, b, and c.
Equation passes at vertex --> 3 = (4)a + 2b + c (1)
y-intercept is zero, then c = 0 (2)
x- intercept is zero, --> 0 = 16a + 4b (3)
Solve system:
(1) --> 3 = 4a + 2b --> b = (3 - 4a)/2
(3) --> 16a + 4b = 0 --> 16a + 6 - 8a = 0 --> 8a = -6 --> a = -3/4.
b = (3 + 3)/2 = 3
Equation: $y = - \frac{3 {x}^{2}}{4} + 3 x$
Check.
x = 0 --> y = 0 .OK
x = 4 --> y = -12 + 12 = 0. OK