question 01: describe the transformations from the parent function, ( y = sin(x) ):\na) ( y = 2 sin(-x)…

Explanation:

Step1: Analyze (y = 2\sin(-x))

For (y = A\sin(Bx - C)+D), here (A = 2), (B=- 1), (C = 0), (D = 0). The amplitude is (|A|=2). Since (B=-1), (y=\sin(x)) is reflected about the (y) - axis (because (y = f(-x)) is a reflection of (y = f(x)) about the (y) - axis) and vertically stretched by a factor of (2) (because (|A|>1)).

Step2: Analyze (y=-4\sin(x + 2))

For (y = A\sin(Bx - C)+D), here (A=-4), (B = 1), (C=-2), (D = 0). The amplitude is (|A| = 4). Since (A=-4), (y=\sin(x)) is reflected about the (x) - axis (because (y=-f(x)) is a reflection of (y = f(x)) about the (x) - axis) and vertically stretched by a factor of (4) (because (|A|>1)). Since (C=-2), using the phase - shift formula (x=\frac{C}{B}), the graph is shifted to the left by (2) units (because (C=-2) and (y=\sin(x - C)) with (C<0) gives (y=\sin(x+|C|))).

Answer:

a) The graph of (y = \sin(x)) is reflected about the (y) - axis and vertically stretched by a factor of (2). d) The graph of (y=\sin(x)) is reflected about the (x) - axis, vertically stretched by a factor of (4), and shifted to the left by (2) units.