Precalculus · Problem 3.1.23

Function transformations are really about tracking how points move—if you know where a few key points on $f$ go, you can see the whole picture shift and stretch. Here, you're applying three layers: a horizontal shift, a vertical stretch and reflection, and a vertical shift. Pick simple points like $(0, 0)$, $(1, 1)$, and $(-1, 1)$ on $f(x) = x^4$, then apply each transformation in order to see exactly where they land on $g(x) = 2 - 3(x-1)^4$. This concrete tracking makes the abstract formula click.

Function transformations are really about tracking how points move—if you know where a few key points on $f$ go, you can see the whole picture shift and stretch. Here, you're applying three layers: a horizontal shift, a vertical stretch and reflection, and a vertical shift. Pick simple points like $(0, 0)$, $(1, 1)$, and $(-1, 1)$ on $f(x) = x^4$, then apply each transformation in order to see exactly where they land on $g(x) = 2 - 3(x-1)^4$. This concrete tracking makes the abstract formula click.