Determining Transformations
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Find the sequence of transformations that \(y = {x^2}\) transforms to \(y = 2{\left( {x + 1} \right)^2} - 3\)
Dilation 2: from \(x\)-axis
Translation 3: \(- ve\) direction \(y\)-axis
Translation 1: \(- ve\) direction \(x\)-axis
Dilation 2: from \(y\)-axis
Translation 3: \(+ ve\) direction \(y\)-axis
Translation 1: \(- ve\) direction \(x\)-axis
Dilation 2: from \(y\)-axis
Translation 3: \(+ ve\) direction \(x\)-axis
Translation 1: \(- ve\) direction \(y\)-axis
Dilation 2: from \(x\)-axis
Translation 3: \(+ ve\) direction \(x\)-axis
Translation 1: \(- ve\) direction \(y\)-axis
Dilation 2: from \(x\)-axis
Translation 3: \(- ve\) direction \(y\)-axis
Translation 1: \(- ve\) direction \(x\)-axis
\begin{align}
&\begin{aligned}
y=x^{2} \rightarrow &\;y^{\prime}=2\left(x^{\prime}+1\right)^{2}-3 \\
& \frac{y^{\prime}+3}{2}=\left(x^{\prime}+1\right)^{2} \\
& \;y=\frac{y^{\prime}+3}{2} \quad x=x^{\prime}+1 \\
& y^{\prime}=2 y-3 \quad x^{\prime}=x-1
\end{aligned}\\
&\text{Dilation of factor 2 from the \(x\)-axis}\\
&\text{Translation of 3 units in negative direction of \(y\)-axis}\\
&\text{Translation of 1 unit in negative direction of \(x\)-axis}\quad \therefore A
\end{align}
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