Question

Describe the function g(x) in terms of f(x) if the graph of g is obtained by vertically stretching f by a factor of 6, then shifting the graph of f to the right by 2 units and upwards by 3 units.

Answer 1

The function g(x) can be described in terms of f(x) as g(x) = 6 * f(x - 2) + 3. This indicates that the function g(x) results from vertically stretching the function f(x) by a factor of 6, shifting it to the right by 2 units, and moving it upwards by 3 units.

Step-by-step explanation:

The function g(x) can be obtained by applying transformations to the function f(x). Starting with f(x), if we vertically stretch it by a factor of 6, it becomes 6f(x). The graph of 6f(x) is vertically stretched compared to the graph of f(x).

Next, shifting the graph of 6f(x) to the right by 2 units is achieved by replacing x in 6f(x) with (x - 2), resulting in 6f(x - 2). This rightward shift moves the entire graph of 6f(x) horizontally to the right by 2 units.

Finally, adding 3 to the function 6f(x - 2) corresponds to moving the graph of 6f(x - 2) upwards by 3 units, giving the function 6f(x - 2) + 3 = g(x). The constant term 3 in the function g(x) represents the upward shift of the graph by 3 units.

Therefore, the function g(x) is expressed as g(x) = 6 * f(x - 2) + 3, denoting that g(x) is the result of vertically stretching f(x) by 6, shifting it to the right by 2 units, and moving it upwards by 3 units, sequentially applying the specified transformations to the original function f(x).