I was confused specifically about what function I'm transforming in this equation

Question

2. Identify the correct order in applying transformations to f(x) to obtain the graph of g(x)=−32f(−2x+1)+2.

a. How do you know those are the correct steps?

Answer 1

horizontal shift left 1

horizontal compression by factor of 1/2

reflection over x axis

vertical stretch by factor 32

reflection over y axis

vertical shift up by 2

6 transformations from f(x) to g(x)

if another post agrees, that's a good clue it's correct

confirmation works, if someone else disagrees, at least

one of us is wrong

Answer 2

Consider transformations from the inside-out.

Assuming f(x) = x, line slope 1, y-intercept 0.

f(2x), above line verticall stretched factor 2

f(-2x), above line reflected over x-axis

f(-2x + 1), above line shifted left 1 unit

32f(-2x + 1), above line vertically stretched 32 units

-32f(-2x + 1), above line reflect over y-axis

-32f(-2x + 1) + 1, above line shifted up 1 unit