What is a equation for this data. X=0,Y=2,X=1,Y=120,and X=2,Y=7200
$$\small{\text{
A parabola
$
y=ax^2+bx+c
$
is a equation. We need to calulate a, b and c.
}}$\\$
\small{\text{
We have $x_1 = 0$ and $y_1 =2 $ and calculate $
y_1=ax_1^2+bx_1+c \Rightarrow 2 = 0 + 0 + c \Rightarrow \textcolor[rgb]{1,0,0}{ c = 2 }
$
}}$\\$
\small{\text{
We also have $x_2 = 1$ and $y_2 =120 $ and calculate $
y_2=ax_2^2+bx_2+2 \Rightarrow 120 = a + b + 2
$
}}
$\\$
\small{\text{
$\Rightarrow a + b = 118 \quad (1)$
}}
$\\$
\small{\text{
Last but not least $x_3 = 2$ and $y_3 =7200 $ and calculate $
y_3=ax_3^2+bx_3+2 \Rightarrow 7200= a*4 + b*2 + 2 $
}}
$\\$
\small{\text{
$\Rightarrow 4a + 2b = 7198 \quad (2)$
}}$\\\\$
(1) \quad a + b = 118 \quad | \quad *2\\
(1) \quad 2a + 2b = 236 \\
(2) \quad 4a + 2b = 7198 \\
(2)-(1): 4a-2a = 2a = 7198 - 236 \\
\textcolor[rgb]{1,0,0}{a = 3481}\\
a+b= 118 \\
3481 + b = 118 \\
b= 118 - 3481 \\
\textcolor[rgb]{1,0,0}{b = -3363}\\
\small{\text{
The equation is \boxed{ $y = 3481x^2 - 3363x+2$ }
}}
$\\\\$
\small{\text{
Proof:
$ y(0) = 0-0+2 = 2 $ okay!
}}$\\$
\small{\text{
Proof:
$ y(1) = 3481-3363+2 = 120 $ okay!
}}$\\$
\small{\text{
Proof:
$ y(2) = 3481*4-3363*2+2 = 7200 $ okay!
}}$$
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