A function f is given and the indicated transformation are applied to its graph (in the given order) write the equation for the final transformed graph f(x)=(x) shift 3 units to the left and shift downward 6 units
To shift the graph of f(x) = x three units to the left, we need to replace x with (x + 3).
Thus, the equation for the graph after the shift is f(x) = (x + 3).
To shift the graph downward six units, we subtract 6 from the function.
Thus, the equation for the final transformed graph is f(x) = (x + 3) - 6, which simplifies to f(x) = x - 3.
To perform each transformation on the function f(x) = x, we can follow these steps:
1. Shift 3 units to the left:
When we shift a function to the left by a certain number of units, we subtract that amount from the x-values. So, to shift f(x) = x 3 units to the left, we substitute x + 3 in place of x:
f(x + 3) = x + 3
2. Shift downward 6 units:
When we shift a function downward by a certain number of units, we subtract that amount from the y-values. So, to shift f(x + 3) = x + 3 downward 6 units, we subtract 6 from the entire function:
f(x + 3) - 6 = x + 3 - 6
f(x + 3) - 6 = x - 3
Therefore, the equation for the final transformed graph is f(x) = x - 3.
To write the equation for the final transformed graph, we need to understand the effects of each transformation and apply them one by one.
1. Shift 3 units to the left:
When a function is shifted to the left by a certain amount, we subtract that amount from the input (x) value. So, to shift f(x) three units to the left, we replace x with (x + 3).
2. Shift downward 6 units:
When a function is shifted downward by a certain amount, we subtract that amount from the output (y) value. So, to shift f(x) downward six units, we subtract 6 from the function itself: f(x) - 6.
Therefore, the equation for the final transformed graph is:
f(x) = (x + 3) - 6.
Simplifying this equation further:
f(x) = x - 3 - 6.
Finally, we can simplify to obtain the simplified equation for the final transformed graph:
f(x) = x - 9.