Exploring parabolas:

 

y = ax+ bx +c

By:Lauren Lee

 

The above equation is known as a quadratic equation.  A quadratic equation is an equation that can be expressed in the form of:

where a is not equal to zero.

Every quadratic equation has U-shaped graph called a parabola.

For my first exploration, I want to start out by looking at a parabola with a fixed a and b constant.Let a=1, b=1, and vary c values.Thus our equation is:y = x≤ + x + c.

 

Letís look at the graph for values c = -1, c = -4, c = 0, c = 1, c = 5.

 

 

 

What is seen in our picture is that as the values of c increase, the parabola narrows and the vertex increases.Similarly, as c decreases, the parabola widens and the vertex decreases.

 

 

Now letís explore what happens when we fix a and c values.Let a = 1, c = 1, and vary b.Thus our equation is:y = x≤ + b x + 1.

 

Letís look at the graph for values b = 0, b = -2, b = -4, b = -7, b = 2, b = 4, b = 7.

 

 

We can see that the 'highest' parabola is x≤ + 0x + 1, which also happens to be the upper limit for the parabolas.As b increases positively, the parabola dips lower and lower to the left. As b decreases negatively, it dips lower to the right.One more thing to notice is that all of the parabolas contain the point (0,1).

 

 

Finally, letís explore what happens when we fix b and c.Let b = 2 and c = 1 and vary the values for a.Thus our equation is:y = ax≤ + x + 1.

 

Letís look at the graph for values a = -1, a = -2, a = -4, a = 0, a = Ĺ, a = 1, a = 2, a = 4.

 

 

We initially notice that when a is positive, the parabola opens upward (concave up).Similarly, when a is negative, the parabola opens downward (concave down).We also see that when a = 0, we are left with the linear equation y = 2x + 1.Once again, all of the parabolas cross at the point (0,1).

Finally, we notice that as |a| increases, the parabola narrows, and as |a| decreases, the parabola widens.

 

Do some more explorations on your own.How does the shape change?How does the position change?

 

 

 

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