using the language of transformations, describe the transformations needed to transform the given function…

using the language of transformations, describe the transformations needed to transform the given function from its parent function\n15. ( f(x)=-2 \tan (x) )\n16. ( g(x)=\frac{1}{2} \tan (2 x) )

using the language of transformations, describe the transformations needed to transform the given function from its parent function\n15. ( f(x)=-2 \tan (x) )\n16. ( g(x)=\frac{1}{2} \tan (2 x) )

Answer

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}).