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**Exploration of Parabolas**

WeÕre going to explore the
equation of a parabola: y=**a**x^{2}+**b**x+**c **for
different values of **a**, **b**, and **c**. First, letÕs look at the graph of a
basic parabola y=x^{2}, where **a**=1, **b**=0, and **c**=0:

Notice the graph opens up,
the vertex is at x=0, and the y-intercept is at y=0. LetÕs vary the value of **a** to determine how the graph changes. LetÕs graph y=x^{2 }(blue),
y=¼x^{2} (green), y=½x^{2 }(purple), y=2x^{2}
(red), and y=4x^{2 }(black) on the same axes.

For all these positive values
of **a**, the graph still opens
up. Notice when 0<**a**<1, the graph appears to be stretched horizontally,
and when **a**>1, the graph appears
to be compressed horizontally.
What happens when **a**<0? Well, we know if **a**=0, then y=0, which is the x-axis, so we can make
assumptions about what the graph will look like when **a** is negative.
We can guess that the graph will reflect about the x-axis when **a** is negative.
LetÕs graph y=x^{2 }(blue), y=-x^{2 }(aqua),
y=-¼x^{2} (green), y=-½x^{2 }(purple), y=-2x^{2}
(red), and y=-4x^{2 }(black) on the same axes.

Notice that our hypothesis is
correct: when -1<**a**<0, the
graph appears to be stretch horizontally below the x-axis, and when** a**<-1, the graph appears to be compressed
horizontally below the x-axis.
Thus, when **a** is negative,
the graph opens down.

Now, letÕs look at different
values of **b** while fixing **a**=1 and **c**=0
in the equation y=**a**x^{2}+**b**x+**c**. First, letÕs graph y=x^{2}+x
(red), where **b**=1, on the same
graph with y=x^{2} (blue):

How does the graph
change? What hypothesis do we want
to make about the effects of the value of **b** on the function? Well, we notice that the graph has shifted to the left
½ and down ¼; however, the shape is not affected. LetÕs explore
further by graphing y=x^{2}+x (red), y=x^{2}-x (black), y=x^{2}+½x
(blue), y=x^{2}-½x (purple), y=x^{2}+3x (aqua), and y=x^{2}-3x
(green) on the same graph.

Notice as **b**
increases from 0, the vertex shifts to the left and down from the origin; as **b** decreases from 0, the vertex shifts to the right
and down from the origin. The
graphs with negative values of **b**
are just reflections about the y=axis of the graphs with positive values of
x. When **b**=1, the vertex is at x= -½ and when **b**=-1, the vertex is at x=½; when **b**=½, the vertex is at x=-¼ and when **b**=-½, the vertex is at x=¼; when **b**=3, the vertex is at x=-1.5 and when **b**=-3, the vertex is at x=1.5. From this, it appears that the vertex
is at x=-**b**/2. LetÕs explore further to see if our
hypothesis is correct.

What happens when we graph y=2x^{2}+**b**x for different values of **b**? LetÕs
graph y=2x^{2}-x (red) and y=2x^{2}-4x (blue):

The vertex when **a**=2
and **b**=-1 is at x=¼, and
the vertex when **a**=2 and **b**=-4 is x=1.
Now, we see that our hypothesis is not completely correct, for –**b**/2 for the first graph would give us ½, but
we have ¼. Instead, it must
be –**b**/2**a** since **a**=1
on our previous graphs and that works here as well.

Finally, letÕs see what happens to the graph when we change
our value of **c**. LetÕs graph y=x^{2}+x+**c** for different values of **c** (note: our **a**=**b**=1). Here is the graph of y=x^{2}-x
(blue), x^{2}-x+2 (red), x^{2}-x+ ½ (green), x^{2}-x-3
(purple):

Notice that the **c** is
the y-intercept of each graph.

What do you think the graph of y=-¼x^{2}+3x-1
will look like? Well from our
discoveries, we can say that the graph will open down since **a** is negative.
We also know that the graph will be fairly wide since -1<**a**<0.
The vertex will be at x=-3/(2*(-¼))=6, so the point is at
(6,8). The y-intercept will be at
y=-1. LetÕs graph to check.

Yes! That is
correct. I think we now have a
pretty good understanding of parabolas.
For a recap, when **a** is positive,
the graph opens up. When **a** is negative, the graph opens down. When |**a**|<1, then the graph stretches horizontally; when
|**a**|>1, the graph compresses
horizontally. The value of **b** helps us determine our vertex. The x-coordinate of our vertex is x=-**b**/2**a**. Finally, the value of **c** gives us the y-intercept.

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