Question

use the given points to find the equation of the line of best fit (regression equation). enter your answer as y=ax^2+bx+c

Explanation:

Step1: Substitute the points into the quadratic equation

We have the quadratic equation \(y = ax^{2}+bx + c\).
Substituting \((-2,5)\): \(5=a(-2)^{2}+b(-2)+c\), which simplifies to \(4a-2b + c=5\).
Substituting \((0,0)\): \(0=a(0)^{2}+b(0)+c\), so \(c = 0\).
Substituting \((2,7)\): \(7=a(2)^{2}+b(2)+c\), which simplifies to \(4a + 2b + c=7\).

Step2: Solve the system of equations

Since \(c = 0\), our system becomes:
\(

$$\begin{cases}4a-2b=5\\4a + 2b=7\end{cases}$$

\)
Add the two equations: \((4a-2b)+(4a + 2b)=5 + 7\)
\(8a=12\), so \(a=\frac{12}{8}=\frac{3}{2}\)
Substitute \(a=\frac{3}{2}\) into \(4a-2b=5\): \(4\times\frac{3}{2}-2b=5\)
\(6-2b=5\), \(2b=1\), \(b=\frac{1}{2}\)

Answer:

\(y=\frac{3}{2}x^{2}+\frac{1}{2}x+0\) or \(y=\frac{3}{2}x^{2}+\frac{1}{2}x\)