# What is the equation, in standard form, of a parabola that contains the following points (–2, 18), (0, 2), (4, 42)?

Jun 13, 2016

$y = 3 {x}^{2} - 2 x + 2$

#### Explanation:

Standard form of equation of a parabola is $y = a {x}^{2} + b x + c$

As it passes through points $\left(- 2 , 18\right)$, $\left(0 , 2\right)$ and $\left(4 , 42\right)$, each of these points satisfies the equation of parabola and hence

$18 = a \cdot 4 + b \cdot \left(- 2\right) + c$ or $4 a - 2 b + c = 18$ ........(A)
$2 = c$ ........(B)
and $42 = a \cdot 16 + b \cdot 4 + c$ or $16 a + 4 b + c = 42$ ........(C)

Now putting (B) in (A) and (C), we get\

$4 a - 2 b = 16$ or $2 a - b = 8$ and .........(1)

$16 a + 4 b = 40$ or $4 a + b = 10$ .........(2)

Adding (1) and (2), we get $6 a = 18$ or $a = 3$

and hence $b = 2 \cdot 3 - 8 = - 2$

Hence equation of parabola is

$y = 3 {x}^{2} - 2 x + 2$ and it appears as shown below

graph{3x^2-2x+2 [-10.21, 9.79, -1.28, 8.72]}