Parabola Explorations

by

Susan Sexton

 

Construct graphs for y = ax2 + bx + c where 2 values of a, b, or c are fixed.

 

 

I.  I will begin by fixing a and b and varying c where a = 1 and b = 3.

 

 

This graph has c = 0

 

This graph has c = 1 and (comparing it to the previous graph)

its y-intercept has also moved from 0 to 1.

 

 

 

Now the c-value is 2 and the y-intercept of t

he corresponding graph is also 2.

 

 

 

 

What would happen if c has a negative value?

 

 

 

 

The graphÕs corresponding y-intercept is

the c-value (whether negative, positive or 0).

 

 

LetÕs put it all on one graph:

 

 

 

 

What would the graph of y = x2 + 3x – 5 be?

 

 

 

 

II.  Next I will fix the values for a and c and choose a variety of values for b.

 

LetÕs start with b = 0.

 

 

The vertex of the parabola lies on the y-axis.

 

 

      

 

 

 

 

The graph has shifted to the left 1 unit compared to y = x2 + 0x +2.    

 

 

 

 

 

 

This graph we saw earlier, but it has shifted to

left 1.5 units compared to y = x2 + 0x +2.

 

 

 

 

 

 

 

This graph shifted one-half unit to the

right compared to y = x2 + 0x +2.

 

 

 

 

 

 

LetÕs put it all on one graph:

 

 

 

 

What would the graph of y = x2 - 4x + 2 be?

 

 

 

 

 

 

 

 

III.  Finally I will fix the values for b and c and choose a variety of values for a.

 

If I choose a = 0 the graph will no longer be a parabola! 

So I will start with a = 1 in the familiar graph of y = x2 + 3x + 2.

 

 

 

 

 

 

 

 

This graph has shifted to the right compared to y = x2 + 3x + 2

but that seems anti-intuitive to what was explored earlier. 

Perhaps looking at the overall shape would tell us something more.

The graphÕs shape is stretched compared to the shape of y = x2 + 3x + 2. 

 

 

 

 

 

 

This graphÕs shape appears to be stretched compared to the graph of

y = 2x2 + 3x + 2 and even more so than the graph of y = x2 + 3x + 2.

 

 

 

                                                                              

 

 

 

 

Compared to the graph of y = x2 + 3x + 2

this graph appears to be compressed.

 

 

 

 

This graphÕs shape is even more compressed as

compared to the shape of y = x2 + 3x + 2.  

 

 

                                                                                                 

 

                                                                                         

LetÕs see these on one graph:

 

 

 

                                                                                                                                                                                                                                                                                                             

What would happen if ÒaÓ had a negative value?

 

LetÕs try it for the graphs graphed above.

 

The graphs all open down when the

value of ÒaÓ is a negative number.

 

       

 

 

                                                                       

Now you try a couple:

Given y = - 4x2 + 2x + 1 can you tell what the graph would look like? (graph)

 

 

Can you give the equation of the graph of the following parabola?

 

(equation)

 

 

 

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