Question

Which Function has the graph below?

Answer 1

C

Explanation:

The general rule for the quadratic function is

`y=ax^2+bx+c`

From the graph you can see that the curve passes through the points (2,4), (1,7) and (3,7), so

`y(2)=4\Rightarrow 4=a\cdot 2^2+b\cdot 2+c\\ \\y(1)=7\Rightarrow 7=a\cdot 1^2+b\cdot 1+c\\ \\y(3)=7\Rightarrow 7=a\cdot 3^2+b\cdot 3+c`

We get the system of three equations:

`\left\{\begin{array}{l}4a+2b+c=4\\ \\a+b+c=7\\ \\9a+3b+c=7\end{array}\right.`

Subtract these equations:

`\left\{\begin{array}{l}4a+2b+c-a-b-c=4-7\\ \\9a+3b+c-a-b-c=7-7\end{array}\right.\Rightarrow \left\{\begin{array}{l}3a+b=-3\\ \\8a+2b=0\end{array}\right.`

From the second equation:

`b=-4a`

Substitute it into the first equation:

`3a-4a=-3\\ \\a=3`

So,

`b=-4\cdot 3=-12`

and

`3+(-12)+c=7\\ \\c=7+9=16`

The quadratic function is

`y=3\cdot x^2-12x+16`