I am teaching a class on precalculus this semester, and I'm having a lot of difficulty with the online homework program we are using on a particular section. We're in the section where we are learning to use parent function transformations, and the homework asks the students to find the equation of the graph. In green is the parent function, and in blue is the transformed graph.
I taught students to solve the problem by using an outline equation and picking a point on the line like so:
$$y = a(x - b)^3 + c,$$ where $b = 1$, $c=2$, and a point that looks like it's on the line is $(-3,-5)$. So we just solve for $a$ to get our vertical stretch/compression:
$$-5 = a(-3 - 1)^3 + 2 \implies a = {7\over 64}.$$ This gives the equation of a graph that is ALMOST correct, but unfortunately the online program doesn't take the answer.
To me, this implies that the point we selected isn't exactly on the line--that the graph ALMOST passes through the point, but the zoom of the graph doesn't allow me to really see if I'm right.
I need to ask if there is anything wrong with the method I am using to teach students for this kind of problem. In other words, am I forgetting something about transformations such as another variable that would help students get the correct answer? I know there should be a horizontal stretch/compression variable, but that doesn't matter here as far as I'm aware.