### Investigation of the 'Graph of the Parabola'

### By Moni Olubuyide

**4. Graph the parabola: y = 2(x-squared) + 3x - 4**

**i. Overlay a new graph replacing each x by (x - 4).**

*ii. Change the equation to move the vertex of the graph into the second quadrant.*

**iii. Change the equation to produce a graph concave down that shares the same vertex.**

iv. Generalize . . .

**1. When you replace each x by (x - 4) that shifted the original graph 4 units to the right.**

*2. Note that in the original equation a = 2, b = 3, and c = -4. The c was the variable that shifted the original triangle down. Therefore to move the vertex of the graph into the second quadrant, I made c = +4. This would have shifted the graph up into the second quadrant. Instead of using the original triangle though, I used the overlay with (x-4). So to move this graph into the second quadrant, I had to shift the graph to the left instead of the right so therefore I changed it to (x +4).*

3.** To change the equation to produce a graph concave down that shares the same vertex, you have to do the opposite of the graph. Therefore a = -2, b= -3 and we would expect c = 4. But c is a little different.. c is equal to -6.25 because to complete the square of the opposite graph we have to take (b/a) and square it then** add 4 which gives us 6.25.