Exercise 24 Page 966

We are given the graph of y=x^2.

We want to use transformations to graph y= - x^2 -3. To do so, we will consider two possible transformations.

1. Reflections
2. Vertical translations

   Let's consider them one at a 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.

   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, 3 is subtracted from the whole function, so the previous graph will be translated three units down.

   

   Final Graph

   Let's now graph the given function.

   

   Finally, let's summarize the transformations that we have used to graph y= - x^2 - 3 starting with the parent function, y=x^2.

   • Reflection across the x-axis
   • Translation three units down