Question

When f(x) becomes f(x - 1) + 2, how does it affect the y-intercept, regions where the graph is increasing and decreasing, and the end behavior?

A) Y-intercept is shifted up by 2 units; regions of increase and decrease remain the same; end behavior is not affected.
B) Y-intercept is shifted down by 2 units; regions of increase and decrease remain the same; end behavior is not affected.
C) Y-intercept remains the same; regions of increase and decrease are shifted right by 1 unit; end behavior is not affected.
D) Y-intercept remains the same; regions of increase and decrease are shifted left by 1 unit; end behavior is not affected.

Answer 1

The transformation of f(x) to f(x - 1) + 2 causes the y-intercept to shift up by 2 units, while the regions of increase and decrease as well as the end behavior of the graph are not affected.

Step-by-step explanation:

When the function f(x) is transformed to f(x - 1) + 2, this modifies the graph of the function in a couple of ways. The subtraction by 1 inside the function shifts the entire graph to the right by 1 unit, which means the regions where the graph is increasing and decreasing are also shifted to the right by 1 unit. However, since this shift is horizontal, it does not alter the end behavior of the function.

Adding 2 to the function after the transformation indicates a vertical shift upwards by 2 units. This means that the y-intercept of the original function, whatever it was, will also be increased by 2. Consequently, the correct option that describes the effect of transforming f(x) to f(x - 1) + 2 is: (A) Y-intercept is shifted up by 2 units; regions of increase and decrease remain the same; end behavior is not affected.