Alex Szatkowski
In this exploration, I will investigate the relationship between the two standard forms of the graph of a parabola.
and
What I know about
- "a" affects the shape (wide, or narrow)
- If a > 0, the graph has an absolute minimum. If a < 0, then the parabola has an absolute maximum
- the vertex can be found by 
- a large "a" will result in a very thin parabola and a very small "a" will result in a very wide parabola
- in order to find the zeros of the function, you can factor and set the factors equal to zero
- a parabola will hit the x axis in 0,1 or 2 places which we can find by factoring
What I know about 
- h determines a horizontal transformation
-if h is negative, there is shift left and if h is positive then there is a shift right
-k determines a vertical transformation
-if k is positive there is a shift up and if k is negative there is a shift down
The first form of a parabola tells me the shape, zeros, minimum and maximum points, and how many times the parabola will hit the x-axis. The second form provides the transformations of the graph of the parabola in a more obvious manner.
Since both forms of the function represent parabolas, I wanted to look at them algebraically to see the relationship that must exist.
First I looked at
Then expanded 
and distributed the two factors ( x - h ) 
Next I distributed "a" 
Finally, I grouped the constants together 
Where 
and 
and 
As we can see, all that needed to be done was expand the equation and do simple distribution and grouping.
Conversely,
Beginning with
Set the equation equal to zero: 
Divide through by a: 
Subtract 
from both sides: 
Using completing the square, add the square of "b" to both sides: 
Factor the right side: 
Bring everything to right side: 
Multiply back through with a: 
We can now see that 
and 
This process was a bit more complex, however I did not need to use anything that students in an algebra course would not know. The most important thing to remember is to not be overwhelmed by all the letters. We must remember that a, b, c, h, and k are just numbers, and we only have one true variable, x.