Exercise 6 Page 50

We want to describe how to transform the parent function y=x^2 to the graph of the given quadratic function. y=-(x+3)^2+2 To do so, we need to consider three possible transformations.

1. Reflections
2. Horizontal translations
3. Vertical translations

   Let's consider them one at the time.

   Reflection

   Whenever x^2 is multiplied by a negative number, we will start by reflecting the graph across the x-axis.

   

   Note how each x-coordinate stays the same, and how each y-coordinate changes its sign.

   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. Here, 3 is being added to x, so the previous graph will be translated 3 units to the left.

   

   Vertical Translation

   If an addition or subtraction is applied to the whole function, the graph will be vertically translated. In the case of addition, the graph will be translated up. In the case of subtraction, it will be moved downwards. In the given equation, 2 is added to the whole function, so the previous graph will be translated 2 units up.

   

   Final graph

   Let's now graph the given function g and the parent function f on the same coordinate grid.

   

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

   • A reflection in the x-axis
   • A horizontal translation 3 units to the left
   • A vertical translation 2 units up