Find two quadratic functions, one that opens upward and one that opens down

Question

Given the x intercepts (-4,0) and (4,0) find where they open upward and downward.

Answer 1

Given the x intercept (-4,0) we get the equation x = -4 which gives us the factor (x + 4) when we add 4 to both sides. Given the x intercept (4,0) we get the equation x = 4 which gives us the factor (x - 4) when we subtract 4 from both sides.

Then we set the factors equal to zero to find the original quadratic equation.

(x + 4) (x - 4) = 0

This yields the following quadratic equation after FOILing:

x² - 16 = 0

The quadratic function that opens up is f(x) = x² - 16

To find the equation that opens down we multiply both sides of the equation by negative 1 which yields:

-x² + 16 = 0

The quadratic function that opens down is f(x) = -x²+ 16

The axis of symmetry (the place where each equation opens upward or downward) is determined by the following equation:

x = -b / 2a (ax² + bx + c = 0 / the standard form of a quadratic equation)

For our problem b = 0 and a = 1 from the first equation.

x =-0 / 2(1) = 0

x = 0 is the axis of symmetry (the place where both equations open up or down).