Outcome: C32 demonstrate an understanding of how the parameter changes affect the graphs of quadratic functions

Originator:  Drew

Explanation:  When the parameters (or variables) of a quadratic equation change, the graph of that equation is always affected in one way or another.  The parameters in a quadratic equation  y= ax^2 + bx + c  are "a", "b" and "c". When any one of these parameters is altered it has an immediate affect on the position or shape of the paraboloa  in the graph. When the slope ("a") is changed, the parabola will either become narrower or wider, and can be flipped with the vertex either at the top or the bottom. If "a" becomes a negative number the vertex will be the maximum point of the parabola, but if "a" is a positive number, the vertex will be the minimum point. A normal slope is 1, the higher this value is, the wider the parabola will open and as it gets lower the parabola will become narrower. When the variable "b" is changed, the horizontal translation, or x-intercept, is also changed. The greater "b" becomes, the greater the value of the x-intercept becomes. This same change applies to the variable "c" as well, the greater the value the greater the translation. The only difference being that "c" is the vertical translation, or y-intercept, rather than the x-intercept. So the y-intercept changes according to the value of "c" and the x-intercept changes according to the value of "b". Another aspect that you have to take into consideration when observing "b" and "c" is that these two variables can affect each other directly. A change in the "b" value also changes, in addition to the x-intercept(s), the x co-ordinate of the vertex and the y-intercept. A change in the "c" value changes not only the y-intercept, but the y co-ordinate of the vertex and the x-intercept(s). 

Examples: 

-->Let's see what transformations this basic quadratic graph (y=x^2) goes through as we alter its parameters.

-->If you change the "b" value you will notice that the x-intercept changes as well as the y-intercept. In this case I changed the x-intercept to -2, and knowing that "h", or the horizontal translation, is equal to half of "b" gives me the value of "b" which is -4. Since the x-intercept/x co-ordinate of the vertex changed to -2, the y-intercept changed as well, to +4, to accomodate the parabola for the change in "b". The quadratic equation for this graph is now y=x^2-4x+4.                                                         

-->This time I changed the y-intercept, or "c" value of my equation  from +4 to +5. As you can see not only did the y-intercept change due to the change in "c", but the y co-ordinate of the vertex also changed , from 0 to 1. This change in "c" also had an affect on the x-intercept, and now instead of being +2 the x-intercept is a complex number. The quadratic equation for this graph is now y=x^2-4x+5.                                                                                             

-->The last parameter change to demonstrate is what changes occurr when the "a" value or the stretch is altered. In this graph you can see that the parabola has been flipped and this is due to a negative stretch. When the "a" value is changed to a negative number, the parabola opens downwards and when it is a positive number, it opens upward. Here the stretch has been changed to -1. This means that the stretch is still a normal quadratic stretch, x^2, only its negative. If the stretch were a Negative number different than one, the parabola would opens downwards, but for it would be narrower and you stretch would be the number (2 for example) times your normal quadratic stretch, 2x^2.You'll also notic too that when the stretch changes it affects the y-intercept, and in this case the y-intercept changes from =5 to -3. After this last parameter change, the quadratic equation for this graph is now y=-x^2+4x-3.