Question

Find the quadratic function that fits the following data. which one function fits.

Answer 1

C

Explanation:

The general rule for the quadratic function is

`y=ax^2+bx+c`

Use the data from the table:

`y(50)=130\Rightarrow 130=a\cdot 50^2+b\cdot 50+c\\ \\y(70)=130\Rightarrow 130=a\cdot 70^2+b\cdot 70+c\\ \\y(90)=200\Rightarrow 200=a\cdot 90^2+b\cdot 90+c`

We get the system of three equations:

`\left\{\begin{array}{l}2500a+50b+c=130\\ \\4900a+70b+c=130\\ \\8100a+90b+c=200\end{array}\right.`

Subtract these equations:

`\left\{\begin{array}{l}4900a+70b+c-2500a-50b-c=130-130\\ \\8100a+90b+c-2500a-50b-c=200-130\end{array}\right.\Rightarrow \left\{\begin{array}{l}2400a+20b=0\\ \\5600a+40b=70\end{array}\right.`

From the first equation

`b=-120a`

Substitute it into the second equation:

`5600a+40\cdot (-120a)=70\Rightarrow 800a=70,\\ \\ a=(7)/(80),\\ \\ b=-120\cdot (7)/(80)=-(21)/(2)=-10.5`

So,

`2500\cdot (7)/(80)+50\cdot (-10.5)+c=130\Rightarrow 218.75-525+c=130\\ \\c=130-218.75+525=436.25`

The quadratic function is

`y=(7)/(80)x^2-10.5x+436.25\\ \\y=0.0875x^2-10.5x+436.25`