Question

The base function ( f(x) = |x| ) is reflected in the x-axis, stretched horizontally by a factor of 2, compressed vertically by a factor of 1/3, and translated 3 units to the left and 5 units down. Write the equation of this transformed function ( g(x) ).

Answer 1

The transformed function g(x) after applying a reflection in the x-axis, horizontal stretch, vertical compression, and translations to the base function f(x) = |x| is g(x) = -(1/3)|x/2 + 3/2| - 5.

Step-by-step explanation:

The student's question involves applying a series of transformations to the base function f(x) = |x|. To achieve the transformed function g(x), we need to follow the sequence of transformations which are reflection, horizontal stretch, vertical compression, and translations. Let's apply these transformations systematically:

• Reflection in the x-axis: f(x) becomes -|x|.
• Horizontal stretch by a factor of 2: Replace x with x/2, which gives us -|x/2|.
• Vertical compression by a factor of 1/3: Multiply the function by 1/3 resulting in -(1/3)|x/2|.
• Translate 3 units to the left: Replace x with (x + 3) to get -(1/3)|x/2 + 3/2|.
• Finally, translate 5 units down: Subtract 5 from the function yielding g(x) = -(1/3)|x/2 + 3/2| - 5.

Therefore, the equation of the transformed function g(x) is g(x) = -(1/3)|x/2 + 3/2| - 5.