Quadratic Transformations
The last section discussed examples of y=ax²+bx+c and all curves had the same basic shape with a minimum or maximum point, and an axis of mirror symmetry. However, it was not possible to relate these features easily to the constants a,b,and c. In this post we will start with y=x² and apply transformations to this curve, so that you can start to relate features of the curve to the constants a,b,and c. Complete the exercises below and all will be clear. The starting point of a transformation is the graph of y=x² shown below:
Vertical Shift
The graph of y=x² can be moved up and down by changing the value of c in the equation y=x²+c. You can discover this for yourself in the next exercise.
Exercise: Vertical Shift of y=x²
The graph of y=x² is shown for reference as the yellow curve and this is a particular case of equation y=x²+c where c=0. Repeat the exercise below a few times to observe how changing c moves the curve y=x²+c.
Changing The Value of c
The interactive graph below shows how the graph can be moved up and down by changing c
Vertical stretch and reflection
The graph of y=ax² can be stretched by changing the value of a; in addition, a negative value of a will reflect the curve along the x-axis.
Exercise: Vertical Stretch of y=x²
The graph of y=x² is shown for reference as the yellow curve and this is a particular case of equation y=ax² where a=1. Repeat the exercise below a few times to observe how changing a stretched and for negative values also reflects the curve y=ax².
The thin blue line is a smooth curve that has been drawn though the points by the computer since the curve is continuous.
Changing The Value of a
The interactive graph below shows how the graph can be stretched and/or reflected by changing a
Multiple Transformations
You can view multiple transformations applied to a simple base quadratic function, in the order: stretch, vertical shift, and finally reflect.
Exercise: Determine transformations
The transformation stretch, vertical shift, and maybe reflect have been applied to the equation
Reflection
Summary
We have discussed the transformations: vertical stretch, vertical shift, and reflect along the x-axis. The only remaining transformation which is horizontal shift, shall be discussed after you have aquired an understanding of factorising a quadratic equation,this shall be examined in detail in the next three sections.