Three functions are given below: f(x), g(x), and h(x). explain how to find the axis of symmetry for each function, and rank the functions based on their axis of symmetry (from smallest to largest). f(x) = –4(x − 8)2 + 3 g(x) = 3x2 + 12x + 15 h(x) = graph of negative 1 times the quantity of x minus 3 squared, plus 2 

The axis of symmetry is a vertical line that cuts a parabola in half. The equation for the axis of symmetry is represented by:

x = -b/2a

We can solve this problem by:

1) Looking at the given graphs of the polynomial functions and looking at where each parabola is cut in half

2) Solving for each axis of symmetry algebraically by first putting each given function into the standard form of ax² + bx + c then solving for the axis of symmetry.

In this problem, I'll be solving for each axis of symmetry algebraically.

The first function: f(x)

f(x) = -4(x - 8)² + 3

f(x) = -4(x - 8)(x - 8) + 3

f(x) = -4(x² - 16x + 64) + 3

f(x) = -4x² + 64x - 256 + 3

f(x) = -4x² + 64x - 253

Axis of symmetry: x = -b / 2a

x = -(64) / 2(-4)

x = -64 / -8

x = 8

Axis of symmetry: x = 8

The second function: g(x)

g(x) = 3x² + 12x + 15

The function function is already in standard form, so just substitute the values into the formula for the axis of symmetry.

x = -b / 2a

x = -(12) / 2(3)

x = -12 / 6

x = -2

Axis of symmetry: x = -2

The third function: h(x)

h(x) = -1(x - 3)² + 2

h(x) = -1(x² - 6x + 9) + 2

h(x) = -x² + 6x - 9 + 2

h(x) = -x² + 6x - 7

Axis of symmetry: x = -b / 2a

x = -b / 2a

x = -(6) / 2(-1)

x = -6 / -2

x = 3

Axis of symmetry: x = 3

Solution: The ranking of the functions based on their axis of symmetry from smallest to largest is g(x), h(x), and f(x).