Transformations: Reflection across the x-axis, and translation one unit up Graph:
We want to describe how to transform the parent function f(x)=x^2 to the graph of the given quadratic function.
g(x)=-x^2+1
To do so, we need to consider four possible transformations.
Notice that we only need to consider the first and the fourth transformation.
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, 1 is added to the whole function, so the previous graph will be translated one unit up.
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 how to draw the graph of the given function when starting with the parent function, f(x)=x^2.
