Sure, let's analyze how the curve with the function y = f(x) is transformed when we apply the given changes:
a) The curve defined by the equation y = f(x + 7):
The function f(x + 7) represents a horizontal shift of the graph of y = f(x). The "+7" inside the function argument means that every x value of the original function is decreased by 7 to produce the same y value in the transformed function. In terms of the graph, this is a shift 7 units to the left.
Since the original vertex of the curve is at (2, -3), shifting the curve left by 7 units moves the vertex to:
(2 - 7, -3) = (-5, -3)
b) The curve defined by the equation y = f(-x):
The function f(-x) represents a reflection of the graph of y = f(x) across the y-axis. This means that for each point (x, y) on the original graph, the transformed function will have a point (-x, y).
Thus, reflecting the vertex (2, -3) across the y-axis changes the x-coordinate from positive to negative, while the y-coordinate remains unaffected. The new coordinates of the vertex are:
(-2, -3)
In summary:
a) The vertex of the curve y = f(x + 7) is at (-5, -3).
b) The vertex of the curve y = f(-x) is at (-2, -3).
