Beginning with the function f(x) = (0.8)x, state what transformations were used on this to obtain the functions given below:

g(x) = - (0.8)x -2
h(x) = ½ (0.8)x-2
k(x) = (0.8)-3x+9

To determine the transformations applied to the original function f(x) = (0.8)x to obtain the given functions g(x), h(x), and k(x), we need to analyze the changes to the equation.

Let's break down each transformation:

1. Function g(x) = - (0.8)x - 2:

- The negative sign, "-", in front of (0.8)x indicates a reflection across the x-axis. This means that the graph of g(x) is obtained by flipping the graph of f(x) upside down.
- The constant term "-2" shifts the graph vertically downward by 2 units. This causes the entire graph of g(x) to move 2 units down compared to f(x).

Therefore, the transformations applied to f(x) to obtain g(x) are a reflection across the x-axis and a vertical shift downward by 2 units.

2. Function h(x) = ½ (0.8)x - 2:

- The coefficient "½" in front of (0.8)x indicates a vertical compression of the graph. This means that each point on the graph of f(x) is vertically halved in height.
- The constant term "-2" shifts the graph vertically downward by 2 units. Similar to g(x), this causes the entire graph of h(x) to move 2 units down compared to f(x).

Therefore, the transformations applied to f(x) to obtain h(x) are a vertical compression by a factor of ½ and a vertical shift downward by 2 units.

3. Function k(x) = (0.8)^(−3x) + 9:

- The exponent "-3x" inside the parentheses indicates a horizontal compression or stretching. This means that the graph of f(x) will be compressed or stretched along the x-axis depending on the value of x.
- The constant term "9" shifts the graph vertically upward by 9 units. This causes the entire graph of k(x) to move 9 units up compared to f(x).

Therefore, the transformations applied to f(x) to obtain k(x) are a horizontal compression or stretching depending on the value of x and a vertical shift upward by 9 units.

By understanding these transformations, we can see how each function is derived from the original function f(x) = (0.8)x.