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Transformations of Quadratic Functions

Transformations of Quadratic Functions 1.6 - Solution

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To determine the quadratic function that is obtained from the parent function f(x)=x2f(x)=x^2 after the given sequence of transformations, let's consider the transformations one at a time.

Horizontal Stretch by a Factor of 22

We'll start by performing a horizontal stretch of the parent function by a factor of 2.2. This is done by multiplying the xx-variable by 12.\frac{1}{2}. The resulting function is y=12x2.y=\frac{1}{2}x^2.

Vertical Translation 22 Units Up

Next, to perform a vertical translation 22 units up we need to add 22 to the whole function. The result is y=12x2+2.y=\frac{1}{2}x^2+2.

Reflection in the yy-axis

Finally, let's reflect the function across the yy-axis by multiplying the variable by -1.\text{-}1. y=12(-1x)2+2\begin{gathered} y=\dfrac{1}{2}\left(\text{-}1\cdot x\right)^2+2 \end{gathered} This will not affect the graph since (-x)2=x2.\left(\text{-} x \right)^2=x^2.

The quadratic function that is obtained after the sequence of transformations is g(x)=12x2+2.g(x)=\frac{1}{2}x^2+2.

Vertex

The vertex of g(x)g(x) let's rewrite it into vertex form. This is done by subtracting 00 from x.x. g(x)=12(x0)2+2\begin{gathered} g(x)=\dfrac{1}{2}(x-{\color{#0000FF}{0}})^2+{\color{#009600}{2}} \end{gathered} Therefore, the vertex of gg is (0,2).({\color{#0000FF}{0}},{\color{#009600}{2}}).