We want to describe how the graph of the given quadratic function is related to the graph of its parent function f(x)=x^2. To do so we will transform the graph of f(x)=x^2 to the graph of the given function.
h(x)=1/2x^2+4
To do so, we need to consider two possible transformations.
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.
In the given exercise, x^2 is multiplied by 12. Therefore, the previous graph will be vertically compressed by a factor of 12.
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, 4 is added to the whole function, so the previous graph will be translated four units 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.
