Question

Suppose the function f(x) is transformed and the new rule for the translated function, g(x), is g(x) = f(x + 6) - 4. Describe this transformation.

Answer 1

The function g(x) = f(x + 6) - 4 represents a horizontal shift of 6 units to the left and a vertical shift of 4 units downward of the original function f(x).

Step-by-step explanation:

When the function f(x) is transformed into the function g(x), the rule g(x) = f(x + 6) - 4 represents two specific transformations. The part where we see x + 6 inside the function indicates a horizontal shift of the graph of f(x) by 6 units to the left. This is because adding a positive number inside the function's argument moves the graph in the negative x-direction. Conversely, subtracting within the argument moves the graph in the positive x-direction.

Additionally, the "- 4" at the end of the function g(x) signifies a vertical shift. The graph of f(x) is moved downward by 4 units. When a constant is added or subtracted to the function outside of its argument, it results in a vertical translation. Specifically, subtracting 4 moves the graph downwards, while adding would move it upwards.

To summarize, the function g(x) = f(x + 6) - 4 is the result of taking the graph of f(x) and shifting it 6 units to the left and 4 units down. These transformations do not affect the shape of the graph; they merely reposition it within the coordinate plane.