EMAT 6680 Assignment 2: Activity 5

Try several graphs of y= ax^2 on the same axes (i.e., use different values of a)

Ashley Jones


Vertical Compression:

In the first animation of the equation y= ax^2 we can see the graph as the parameter a varies as it relates to the parent graph of quadratic functions, y= x^2 (where a=1). In this particular visual a oscillates between the values of 0 and 1. The quadratic equation itself represents a parabola. Since this illustration shows the graph with just a varying between 0 and 1, then we can determine that only the width of the parabola is affected. The vertex remains the same, located at the origin. In addition, because the value of a stays positive in this example, the parabola is always opening up. While a varies between 0 and 1 we notice that the width becomes much wider than the graph of y= x^2. This is because when the absolute value of a is less than 1, we have a vertical compression of the parabola. A vertical compression 'pushes' the parabola to be wider, thus compressing it in the vertical direction.

Varying parameter a in graph: y= ax^2 :

Vertical Compression of Parabola


Vertical Compression With Reflection:

In this illustration we have the same vertical compression occurring, except the parabola is now opening downward. This change in the graph is due to the fact that we are now varying the parameter a between the values of -1 and 0. While we take the absolute value of a to determine if we have a vertical compression, we still must consider if a is a positive value or not. As in this example, if a is negative, our parabola opens downward. This negative value affects the graph by reflecting it across the x-axis.

Varying parameter a in graph: y= ax^2 :

Vertical Compression and Reflection of Parabola

Vertical Stretch:

The next set of visuals show us the graph of y= ax^2 as a varies between 1 and 10 (graph on top) and -10 and -1 (graph on bottom). As we can tell, in both examples the width of the parabola is decreasing. This is called a vertical stretch. A vertical stretch 'pulls' the parabola more narrow, thus stretching it in the upward direction. The top graph animation opens up since the parameter a remains positive as it oscillates, and the bottom graph animation opens down since the parameter a remains negative as it oscillates.

Varying parameter a in graph: y= ax^2 :

Vertical Stretch of a Parabola

Vertical Stretch and Reflection of a Parabola