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## Transformations of Quadratic Functions 1.6 - Solution

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 $\text{-}1.$ $\begin{gathered} y=\dfrac{1}{2}\left(\text{-}1\cdot x\right)^2+2 \end{gathered}$ 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 $x.$ $\begin{gathered} g(x)=\dfrac{1}{2}(x-{\color{#0000FF}{0}})^2+{\color{#009600}{2}} \end{gathered}$ Therefore, the vertex of $g$ is $({\color{#0000FF}{0}},{\color{#009600}{2}}).$