Describing Transformations of Quadratic Functions

To determine the quadratic function that is obtained from the parent function $f(x)=x^2$ after the given sequence of transformations, let's consider the transformations one at a time.

Horizontal Stretch by a Factor of $2$

We'll start by performing a horizontal stretch of the parent function by a factor of $2.$ This is done by multiplying the $x$-variable by $\frac{1}{2}.$ The resulting function is $y=\frac{1}{2}{x}^2.$

Vertical Translation $2$ Units Up

Next, to perform a vertical translation $2$ units up we need to add $2$ to the whole function. The result is $y=\frac{1}{2}{x}^2+2.$

Reflection in the $y$-axis

Finally, let's reflect the function across the $y$-axis by multiplying the variable by $\text{-}1.$ $$\begin{array}{c}y=\frac{1}{2}{\left(\text{-}1\cdot x\right)}^2+2\end{array}$$ This will not affect the graph since ${\left(\text{-}x\right)}^2=x^2.$

The quadratic function that is obtained after the sequence of transformations is $g(x)=\frac{1}{2}{x}^2+2.$

Vertex

The vertex of $g(x)$ let's rewrite it into vertex form. This is done by subtracting $0$ from $x.$ $$\begin{array}{c}g(x)=\frac{1}{2}(x-0{)}^2+2\end{array}$$ Therefore, the vertex of $g$ is $(0,2).$