Resources for Translations, Reflections and Dilations
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Questions
5
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Video Tutorials
2
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Translations, Reflections and Dilations Theory
![For a reflection about the \(x\)-axis \[ y=f(x) \rightarrow y=-f(x) \] For a reflection about the \(y\)-axis \[ y=f(x) \rightarrow y=f(-x) \] For a vertical dilation of a scale factor \(k\) \[ y=f(x) \rightarrow y=k f(x) \] \columnbreak For a horizontal dilation of a scale factor \(\dfrac{1}{k}\) \[ y=f(x) \rightarrow y=f(k x) \] For a vertical translation for \(k>0\). \[ y=f(x) \rightarrow y=f(x)+k \] For a horizontal translation for \(k>0\) the graph is translate and to the left by \(k\) units \[ y=f(x) \rightarrow y=f(x+k) \] \end{multicols} \textbf{Order of transformations}\\ For \(y=k f(a(x+b))+c\)\\ - do horizontal dilation a then horizontal translation \(b\)\\ - do vertical dilation \(k\), then vertical translation \(c\).\\ \begin{multicols}{2} \textbf{Example 1}\\ \(y=(x-2)^3\) is dilated vertically by a factor of 2 and reflected about the \(x\)-axis. The new function is?\\ \textbf{Example 1 solution}\\ \(y=(x-2)^3 \rightarrow y=2(x-3)^3 \rightarrow y=-2(x-3)^3\)\\ \textbf{Example 2}\\ \(y=x^2+1\) is translated 2 units to the left and 1 unit up. The new function is?\\ \textbf{Example 2 solution}\\ $\begin{aligned} y & =x^2+1 \rightarrow y=(x+2)^2+1 \rightarrow y=(x+2)^2+1+1 \\ \therefore & y=(x+2)^2+2 \end{aligned}$\\ \columnbreak \textbf{Example 3}\\ \(y=x^3+1\) is dilated horizontally by a factor of \(\dfrac{1}{2}\) and translated 1 unit to the right. The new function is?\\ \textbf{Example 3 solution}\\ $\begin{aligned} y & =x^3+1 \rightarrow y=(2 x)^3+1 \rightarrow y=8 x^3+1 \\ & \rightarrow y=8(x-1)^3+1 \end{aligned}$\\ \end{multicols}](/media/pj0ph4xq/4775.png)