Question

using the language of transformations, describe the transformations needed to transform the given function from its parent function

1. ( f(x)=-2 \tan (x) )
2. ( g(x)=\frac{1}{2} \tan (2 x) )

Explanation:

Step1: Analyze the transformation for \(f(x)= - 2\tan(x)\)

For a function \(y = A\tan(x)\), where \(A\) is a non - zero real number. When \(A=-2\), the transformation from the parent function \(y = \tan(x)\) is a vertical stretch by a factor of \(|A| = 2\) and a reflection about the \(x\) - axis (because \(A=-2<0\)).

Step2: Analyze the transformation for \(g(x)=\frac{1}{2}\tan(2x)\)

For a function \(y = B\tan(Cx)\), where \(B\) and \(C\) are non - zero real numbers.
For the vertical transformation: When \(B = \frac{1}{2}\), there is a vertical compression by a factor of \(|B|=\frac{1}{2}\) from the parent function \(y=\tan(x)\).
For the horizontal transformation: When \(C = 2\), using the formula for horizontal compression \(y=\tan(Cx)\) (where \(C>1\)), there is a horizontal compression by a factor of \(\frac{1}{C}=\frac{1}{2}\) from the parent function \(y = \tan(x)\).

Answer:

1. The function \(f(x)=-2\tan(x)\) is obtained from the parent function \(y = \tan(x)\) by a vertical stretch by a factor of \(2\) and a reflection about the \(x\) - axis.
2. The function \(g(x)=\frac{1}{2}\tan(2x)\) is obtained from the parent function \(y=\tan(x)\) by a vertical compression by a factor of \(\frac{1}{2}\) and a horizontal compression by a factor of \(\frac{1}{2}\).