How to get co-relate the curves

Question

I come across a question where I need to co-relate the function $f(t)$ and $g(t)$. The answer is supposed to be $g(t)=f(\frac{t}{2}-\frac{3}{2})$. Can someone explain the steps involved to reach the expression from the plots shown below?

Answer 1

We can think of the graph of $g(t)$ as being a transformation of the graph of $f(t)$ by a sequence of two steps:

1. Stretch the graph of $f(t)$ horizontally by a factor of two, so that the space between the vertical bars is two rather than one. This gives us the graph of $f\left(\frac{t}{2}\right)$.
2. Shift the resulting graph to the right by $3$ so that the left corner point is at $x = 3$ instead of $x = 1$. This gives us the graph of $f\left(\frac{t - 3}{2}\right)$.

We can simplify this transformation as $$f\left(\frac{t - 3}{2}\right) = f\left(\frac{t}{2} - \frac{3}{2}\right).$$