I was graphing the equation $y = x^2 - a$ and I know why the graph is a parabola intersecting at the points $(-1,0)$ and $(1,0)$.
However, when I graph $y = (x^2 - a)^2$, the graph oddly changes, as if two parabolas have subtracted each other. The greater the value of $a$ gets, the further up this change occurs.
I understand that the $a$ is squared, so as it is further increased/decreased away from $0$, the $y$ intercept changes.
From a mathematical perspective, how can this be explained further - why it looks like the subtracting each other?
Thanks
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If $a>0$ is the same in the two functions, the correct graph is this.
Note that $y=x^2-a$ is a parabola that has negative values $y<0$ in the interval $x \in(-\sqrt{a},\sqrt{a})$.
For the squared equation $y=(x^2-a)^2$ we can have only positive values, so the graph become positive in the interval $(-\sqrt{a},\sqrt{a})$, but we have always $y=0$ for $x=\pm \sqrt{a}$, so these values are roots of the function and are also local minima. They are double roots i.e. values in which the function an its derivative are $0$.
Also note that the intercept at $x=0$ is simply the square of $a$.