পরাবৃত্তের সমীকরণ ($y=ax^2$)

The parabola with the equation $y=ax^2$ is very similar to its parent function (the function without any modifications): $y=x^2$.

First we have to understand what the coefficient $a$ does to the parent function:

Let us write down some values for the parabola and parent function.

$y=x^2: \ \dots;(-3,9);(-2,4);(-1,1);(0,0);(1,1);(2,4);(3,9);\dots \

y=ax^2: \ \dots;(-3,9a);(-2,4a);(-1,1a);(0,0);(1,1a);(2,4a);(3,9a);\dots$

As you can see, $y=ax^2$ is simply has a $y$ value $a$ times more than $y=x^2$. In a more graphical sense, the graph of the parabola is the parent function stretched.

The green plot is the parent function $y=x^2$ while the blue function is the parabola with $a=5$. Note how the vertex is the same for both functions.