Exploring Parabolas

By

Princess Browne

Try several graphs of y = ax2

On the same axes (i.e. use different values of a)

I will start by graphing the equation of y = x2

The graph of y = x2 is a parabola that is concave upward.

LetŐs look at few more graphs where a is greater than 1.

Purple

Orange

Blue

Green

Teal

From the graphs above, we can see that the graph is positive or concave upward when a is positive. Each parabola share the same vertex point. The graph gets skinnier or closer to the y-axis as a increases. The parabola does not have a constant slope. If we look at the graph  of y= x we will notice that as x increases by 1, starting with x = 0, y increase by 1, 3, 5, 7, É As  x decreases by 1 starting with x = 0, y still increases by 1, 3, 5, 7,É Look at the table below for better understanding.

 X 1 2 3 4 y 1 4 9 16

Find slopes using (1,1), (2,4), (3,9) and (4,16)

(y2-y1)/(x2-x1) = (4-1)/ (2-1) = 3

(y4-y3)/(x4-x3) = (19-9)/(4-3) = 7

Using the models above we will predict that when a is less than 1 the graph will get wider than the original graph. LetŐs see our prediction of the graph when a is less than 1.

Purple

Red

Blue

Green

When a is less than 1 the graph is larger than the graph of y=x2 and the points are moving away from the vertex. The graph is closer to the x-axis and the graph is always a parabola. Next, letŐs see what will happen to the graph when a is a negative number.

Purple   y = -x2

Orange  y = -2x2

Blue      y = -3x2

Green    y = -4x2

Teal       y = -5x2

The graph is still a parabola. The parabola is negative or concave downward symmetric to the y-axis. As a increases the graph gets skinnier. The graph is concave up (along the y-axis) when a is positive and concave down (along the y-axis) when a is negative. The positive graph has a minimum point; whereas the negative graph has a maximum point. Now we will add a constant c to the equation and see what will happen to the graph. The graph below is the graph of ax2 + c.

Purple

Orange

Blue

Green

Adding or subtracting the constant c will shift the graph along the y-axis. When c is added to the equation the graph is above the x-axis and moves up along the y-axis. When c is subtracted from the equation the graph is below the x-axis and moves down along the y-axis. For example: when c is 2, the graph will move up 2 units from the graph of y = x2. The vertex of y = x2 is (0,0) and the vertex of y = x2+ 2 is (0,2).

Finally, we will compare the graph a(x-b)2 + c with the graph of ax2 + c.

Teal                y = x2 +1

Purple            y = (x-2)2 + 1

Red                y = (x-3)2 + 2

Blue               y = (x+2)2 + 1

Green             y = (x+3)2 + 2

The graph of y = (x-2)2 + 1 will shift the graph two units right from the graph of y = x2 +1 and the vertex is (2,1). The graph of y = (x+2)2 + 1 will shift the graph two units left from the graph of y = x2 +1 and the vertex is (-2,1). The equation of y = x2 +1 will shift the parabola up one unit from the vertex or along the y-axis. Whereas, the equation of y = (x-2)2 will shift the parabola two units right of the vertex or along the x-axis.