Account for order in two horizontal transformations

The Teacher completes the intermediate expression and emphasises: replace x by $2x$, then replace x by $x+1/2$. Substituting into the entire current input gives $2(x+1/2)$, not $2x+1/2$.

Scale first

$$y=f(x)\quad\longrightarrow\quad y=f(2x)$$

Halve each horizontal distance from the y-axis.

Translate left by one half

$$\begin{aligned}&y\\ & =f\!\left(2\left(x+\frac12\right)\right)\\ & =f(2x+1)\end{aligned}$$

Apply the replacement to the current equation before simplifying.

The Teacher supplies both transformations and stresses the replacements: x by $x+1$, then x by $2x$. The translation is one unit before scaling, rather than one half after scaling.

Translate left by one

$$y=f(x+1)$$

Replace x by $x+1$ in the original graph.

Scale horizontally

$$y=f(2x+1)$$

Replace x by $2x$ in that equation; the constant 1 remains.