Question

Write a function using the format f(x)=a(bx-h)+k that transforms the parent function through a horizontal shrink by a factor of 1/3, a horizontal translation 6 units to the right, followed by a vertical compression (shrink) by a factor of 0.5, a reflection in the x-axis and finally a vertical translation 7 units up.

Answer 1

The transformed function reflecting a series of transformations on the parent function is f(x) = -0.5(3x - 18) + 7, incorporating a horizontal shrink, horizontal translation, vertical compression, reflection, and vertical translation.

Step-by-step explanation:

To describe a transformation of the parent function that includes a horizontal shrink by a factor of 1/3, a horizontal translation 6 units to the right, a vertical compression by a factor of 0.5, a reflection in the x-axis, and a vertical translation 7 units up using the function format f(x) = a(bx-h)+k, we need to apply these transformations step by step.

First, the horizontal shrink by 1/3 is achieved by multiplying x by 3 (since we are shrinking, we divide by the factor, which equates to multiplying the term by its reciprocal).

Next, the horizontal translation 6 units to the right is represented by subtracting 6 from x, which gives us (x - 6). Now, the vertical compression by 0.5 is shown by multiplying the entire function by 0.5. The reflection in the x-axis is achieved by adding a negative sign to this factor, resulting in -0.5. Finally, the vertical translation 7 units up is done by adding 7 to the function.

Therefore, the transformed function is f(x) = -0.5(3x - 18) + 7. This applies the series of transformations in the proper order to the parent function.