Question

find the quadratic function that is the best fit for ( g(x) ) defined by the table below.
the quadratic function is ( y=square x^{2}+square x+square ).
(type integers or decimals.)

Explanation:

Step1: Set up the system of equations

Let the quadratic function be \(y = ax^{2}+bx + c\).
When \(x = 0\), \(y=4\), so \(4=a\times0^{2}+b\times0 + c\), which gives \(c = 4\).
When \(x = 2\), \(y = 3.6\), so \(3.6=a\times(2)^{2}+b\times2+4\), which simplifies to \(4a + 2b=3.6 - 4=- 0.4\) or \(2a + b=-0.2\).
When \(x = 4\), \(y = 6.4\), so \(6.4=a\times(4)^{2}+b\times4 + 4\), which simplifies to \(16a+4b=6.4 - 4 = 2.4\) or \(4a + b = 0.6\).

Step2: Solve the system of equations

Subtract the equation \(2a + b=-0.2\) from \(4a + b = 0.6\).
\((4a + b)-(2a + b)=0.6-(-0.2)\)
\(4a + b - 2a - b=0.6 + 0.2\)
\(2a=0.8\), so \(a = 0.4\).
Substitute \(a = 0.4\) into \(2a + b=-0.2\), we get \(2\times0.4 + b=-0.2\), \(0.8 + b=-0.2\), so \(b=-1\).

Answer:

The quadratic function is \(y = 0.4x^{2}-1x + 4\)