Notice that we only need to consider the first, second, and fourth transformations.
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.
Stretch or Compression
We have a vertical stretch when x^2 is multiplied by a number whose absolute value is greater than one. If x^2 is multiplied by a number whose absolute value is less than one, a vertical compression will take place.
If x^2 is being multiplied by a negative number, the above still applies but everything will be upside down. In the given exercise, notice that the given graph is stretched vertically. x^2 is multiplied by -2. Therefore, the previous graph will be vertically stretched by a factor of 2.
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 exercise, the graph is translated one unit up compared to the previous graph.
Final Graph
Let's now graph the given function and the parent function y=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, y=x^2.
Reflection across the x-axis
Vertical stretch by a scale factor of 2
Translation one unit up
Therefore the function we are looking for is y=-2x^2+1, and the correct answer is D.
