Exercise 24 Page 52

We want to describe how to transform the parent function f(x)=x^2 to the graph of the given quadratic function. y=1/2(x-1)^2 To do so, we need to consider two possible transformations.

1. Vertical stretches and shrinks
2. Horizontal translations

   Let's consider them one at the time.

   Vertical Stretch or Shrink

   We have a vertical stretch when x^2 is multiplied by a number greater than one. If x^2 is multiplied by a number less than one, a vertical shrink will take place.
   
   In the given exercise, x^2 is multiplied by 12. Therefore, the graph of f will be vertically shrunk by a factor of 12.
   

   Horizontal Translation

   If an addition or subtraction is applied to only the x-variable, the graph will be horizontally translated. In case of addition, the graph will be translated to the left. In case of subtraction, it will be moved to the right. In the given equation, 1 is being subtracted from x, so the previous graph will be translated one unit to the right.

   

   Final Graph

   Let's now graph the given function and the parent function f(x)=x^2 on the same coordinate grid.

   

   Finally, let's summarize all the transformations of the graph of f.

   • A vertical shrink by a factor of 12
   • A horizontal translation 1 unit right