Transformations of cosine, logarithm and circle graphs

Problem

The graph of $y=\cos(\pi x)$ is given. a) Apply a translation of $2$ upward and a dilation by a factor of $2$ along the $x$-axis; write the final equation. b) Apply the same transformation (translation and dilation) to $y=\log_2 x$. c) Instead (careful, not the same as before) apply a translation of $2$ to the right, followed by a dilation by a factor of $2$ along $x$, to the unit circle $x^{2}+y^{2}=1$.

Solution

A dilation by a factor of $2$ along $x$ amounts to replacing $x\to \tfrac{x}{2}$. a) Upward translation: $y=\cos(\pi x)+2$; dilation: $y=2+\cos\!\left(\dfrac{\pi x}{2}\right)$. b) Likewise: $y=2+\log_2\!\left(\dfrac{x}{2}\right)$. c) Translation right by $2$: $(x-2)^{2}+y^{2}=1$; then dilation $x\to \tfrac{x}{2}$: $\left(\dfrac{x}{2}-2\right)^{2}+y^{2}=1$, i.e. an ellipse with centre $(4,0)$ and semi-axes $2$ and $1$. $\boxed{\,a)\ y=2+\cos\!\tfrac{\pi x}{2}\,;\quad b)\ y=2+\log_2\!\tfrac{x}{2}\,;\quad c)\ \tfrac{(x-4)^{2}}{4}+y^{2}=1\,}$