The graph of 𝑦=𝑓(𝑥) is reflected in the 𝑥-axis, then horizontally stretched by a factor of 2, and then translated 2 units to the right and 5 unit up. Determine the equation of the final graph.

Answer 1

The rules for transforming a function f(x) into g(x):

g(x) = a·f(b(x-c)) + d

1. a = vertical dilation (stretch); reflects across x-axis if negative
2. b = horizontal dilation (stretch); reflects across y-axis if negative
3. c = horizontal translation (shift); shift right if c is positive, left if c is negative
4. d = vertical translation; up if d is positive, down if d is negative

So for your problem:

1. reflected across x-axis means a = -1
2. horizontally stretched by a factor of 2 means b = 2
3. translated 2 units right means c = 2
4. and 5 units up means d = 5

g(x) = (-1)·f(2(x-2)) + 5 = -f(2x-4) + 5