Step 1 :The quadratic function is generally in the form of \(f(x) = ax^2 + bx + c\). We can use the given points to form a system of equations to solve for a, b, and c.
Step 2 :Given points are \((0,2)\), \((2,-3.6)\), \((4,-8.4)\), and \((6,-12.4)\).
Step 3 :Substituting these points into the equation \(f(x) = ax^2 + bx + c\), we get:
Step 4 :For \((0,2)\): \(2 = c\)
Step 5 :For \((2,-3.6)\): \(-3.6 = 4a + 2b + 2\)
Step 6 :For \((4,-8.4)\): \(-8.4 = 16a + 4b + 2\)
Step 7 :For \((6,-12.4)\): \(-12.4 = 36a + 6b + 2\)
Step 8 :We now have a system of equations:
Step 9 :\[\begin{align*} 1) & c = 2 \\ 2) & 4a + 2b = -3.6 - 2 = -5.6 \\ 3) & 16a + 4b = -8.4 - 2 = -10.4 \\ 4) & 36a + 6b = -12.4 - 2 = -14.4 \end{align*}\]
Step 10 :We can simplify equations 2, 3, and 4 by dividing by 2, 4, and 6 respectively:
Step 11 :\[\begin{align*} 2) & 2a + b = -2.8 \\ 3) & 4a + b = -2.6 \\ 4) & 6a + b = -2.4 \end{align*}\]
Step 12 :Subtracting equation 2 from equation 3, we get \(2a = 0.2\), so \(a = 0.1\).
Step 13 :Substituting \(a = 0.1\) into equation 2, we get \(2*0.1 + b = -2.8\), so \(b = -2.8 - 0.2 = -3\).
Step 14 :So, the quadratic function that best fits the given points is \(\boxed{f(x) = 0.1x^2 - 3x + 2}\).