# MUHAMMET ARICAN

## Assignment 2: Graphing the parabola

In this assignment I am going to introduce the parabolas for some special forms. Parabolas can be written different ways but

in general a parabola has the formula y = ax² + bx+ c here a,b, and c are real numbers.

Here is a graph for parabola y=2x²+3x-4

Here are some methods to examine parabolas:

• Overlaying a new graph replacing each x by (x - 4):

This is the graph of parabola y=2(x-4)² + 3(x-4)-4

As you see if we replace x with (x-4) then the vertex of the graph moves 4 units right on the X axis.

• Changing the equation to move the vertex of the graph into the second quadrant:

As you can see from above graph when we increase the value of the constant c then vertex of the graph moves up.

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

When we change the signs of the parabola's then vertex of the graph inverses.

• Generalization

If a parabola has the general formula y = ax²+bx+c then we can generalize some of important properties:

1. If a>0 then the parabola's concave curve up. If a<0 then the concave curve is down.
2. x = -b/2a gives the minimum or the maximum point of parabola
3. If we give the value 0 for x then we will find the value of Y axis.
4. If we give 0 for y then we will find values for X axis.
5. If we call D=b²-4ac then we can write 3 situations for D :
• If D>0 then parabola and X axis will have two intersection points.
• If D<0 then parabola and X axis will not have intersection points.
• If D=0 then parabola will be tangent to X axis.