A parabola opens upward and has no vertical stretch. The complex roots of the quadratic function are 6 + 4i and 6 – 4i. Determine the function rule.

Describing the function rule means that you are going to write the equation of the parabola using that roots. If x = 6 + 4i, then the factor for that is

(x - 6 - 4i).

If x = 6 - 4i, then the factor for that is

(x - 6 + 4i).

FOILing that together gives you a long string of x- and i-terms with a constant or 2 thrown in:

[tex]x^2-6x+4ix-6x+36-24i-4ix+24i-16i^2[/tex]

What's nice here is that 4ix and -4ix cancel each other out; likewise 24i and -24i. Once that is all canceled away, we are left with

[tex]x^2-12x+36-16i^2[/tex]

The i-squared is what makes this complex. i-squared = -1, so

[tex]x^2-12x+36-16(-1)[/tex] and

[tex]x^2-12x+36+16[/tex] and

[tex]x^2-12x+52=y[/tex]