Question

Describe the steps required to determine the equation of a quadratic function given its zeros and a point.

Answer 1

Procedure:

1) Form a system of 3 linear equations based on the two zeroes and a point.

2) Solve the resulting system by analytical methods.

3) Substitute all coefficients.

Explanation:

A quadratic function is a polynomial of the form:

`y = a\cdot x^(2)+b\cdot x + c` (1)

Where:

`x` - Independent variable.

`y` - Dependent variable.

`a`,
`b`,
`c` - Coefficients.

A value of
`x` is a zero of the quadratic function if and only if
`y = 0`. By Fundamental Theorem of Algebra, quadratic functions with real coefficients may have two real solutions. We know the following three points:
`A(x,y) = (r_(1), 0)`,
`B(x,y) = (r_(2),0)` and
`C(x,y) = (x,y)`

Based on such information, we form the following system of linear equations:

`a\cdot r_(1)^(2)+b\cdot r_(1) + c = 0` (2)

`a\cdot r_(2)^(2)+b\cdot r_(2) + c = 0` (3)

`a\cdot x^(2) + b\cdot x + c = y` (4)

There are several forms of solving the system of equations. We decide to solve for all coefficients by determinants:

`a = \frac{\left|\begin{array}{ccc}0&r_(1)&1\\0&r_(2)&1\\y&x&1\end{array}\right| }{\left|\begin{array}{ccc}r_(1)^(2)&r_(1)&1\\r_(2)^(2)&r_(2)&1\\x^(2)&x&1\end{array}\right| }`

`a = (y\cdot r_(1)-y\cdot r_(2))/(r_(1)^(2)\cdot r_(2)+r_(2)^(2)\cdot x+x^(2)\cdot r_(1)-x^(2)\cdot r_(2)-r_(2)^(2)\cdot r_(1)-r_(1)^(2)\cdot x)`

`a = (y\cdot (r_(1)-r_(2)))/(r_(1)^(2)\cdot r_(2)+r_(2)^(2)\cdot x +x^(2)\cdot r_(1)-x^(2)\cdot r_(2)-r_(2)^(2)\cdot r_(1)-r_(1)^(2)\cdot x)`

`b = \frac{\left|\begin{array}{ccc}r_(1)^(2)&0&1\\r_(2)^(2)&0&1\\x^(2)&y&1\end{array}\right| }{\left|\begin{array}{ccc}r_(1)^(2)&r_(1)&1\\r_(2)^(2)&r_(2)&1\\x^(2)&x&1\end{array}\right| }`

`b = ((r_(2)^(2)-r_(1)^(2))\cdot y)/(r_(1)^(2)\cdot r_(2)+r_(2)^(2)\cdot x +x^(2)\cdot r_(1)-x^(2)\cdot r_(2)-r_(2)^(2)\cdot r_(1)-r_(1)^(2)\cdot x)`

`c = \frac{\left|\begin{array}{ccc}r_(1)^(2)&r_(1)&0\\r_(2)^(2)&r_(2)&0\\x^(2)&x&y\end{array}\right| }{\left|\begin{array}{ccc}r_(1)^(2)&r_(1)&1\\r_(2)^(2)&r_(2)&1\\x^(2)&x&1\end{array}\right| }`

`c = ((r_(1)^(2)\cdot r_(2)-r_(2)^(2)\cdot r_(1))\cdot y)/(r_(1)^(2)\cdot r_(2)+r_(2)^(2)\cdot x + x^(2)\cdot r_(1)-x^(2)\cdot r_(2)-r_(2)^(2)\cdot r_(1)-r_(1)^(2)\cdot x)`

And finally we obtain the equation of the quadratic function given two zeroes and a point.