We can first graph the function $y = a x^{2}$ for the simplest case — this is when $a = 1 .$ Then, we can add graphs of the function considering other values of $a .$ Graphing the functions together will allow us to see the effects of $a .$ We will do this for the different cases mentioned in the exercise.

Case $0 < a < 1$

For this analysis we will study the effects of using $a = 0.25 .$ To do this we can graph $y_{1} = x^{2}$ and $y_{2} = 0.25 x^{2} .$

As we can see, the factor $a = 0.25$ reduces the value of the original function at every point, $shrinking$ the graph vertically and taking it to be closer to the $x - axis .$

Case $a > 1$

For this analysis we can use $a = 2 .$ Therefore, we will graph $y_{1} = x^{2}$ and $y_{2} = 2 x^{2} .$

As we can see, the factor $a = 2$ increases the value of the original function at every point, $stretching$ the graph vertically and taking it away from the $x - axis .$

Case $- 1 < a < 0$

For this case, it will be useful to graph three functions together to visualize all of the transformations taking place. We can graph $y_{1} = x^{2},$ $y_{2} = - x^{2}$ and $y_{3} = - 0.25 x^{2} .$

As we can see, having a negative $a$ parameter causes a reflection in the $x - axis .$ Furthermore, having $a = - 0.25$ decreases in absolute value the $y - values$ of $y_{2} = - x^{2},$ $shrinking$ the graph vertically and taking it closer to the $x - axis .$

Case $- 1 < a$

Once more, it is useful to graph three functions together to visualize all of the transformations taking place. We will graph $y_{1} = x^{2},$ $y_{2} = - x^{2},$ and $y_{3} = - 2 x^{2}$ this time.

As we already mentioned above, having a negative $a$ parameter causes a reflection in the $x - axis .$ However, now that we are using $a = - 2,$ the parameter increases in absolute value the $y - values$ of the function $y_{2} = - x^{2},$ $stretching$ the graph vertically and taking it away from the $x - axis .$

Conclusions

From the results we found above we can summarize the effects of the parameter $a$ in the graph of $y = a x^{2}$

If $a < 0$ it causes a reflection in the $x - axis .$
If $0 < ∣ a ∣ < 1$ it causes a vertical shrink, taking the graph closer to the $x - axis .$
If $1 < ∣ a ∣$ it causes a vertical stretch, taking the graph away from the $x - axis .$

Case 0<a<1

Case a>1

Case -1<a<0

Case -1<a

Conclusions

Case $0 < a < 1$

Case $a > 1$

Case $- 1 < a < 0$

Case $- 1 < a$

Exercise