Assignment 2: Quadratics
Varying Coefficients
Here's a graph of a
quadratic equation
![](../Resources/Assignment2/Quadratic1.gif)
![](../Resources/Assignment2/1.gif)
What happens if we adjust the
term?
The following graph the
adds the quadratic equation
:
![](../Resources/Assignment2/2.gif)
Blue:
,
Purple: ![](../Resources/Assignment2/Quad2Small.gif)
What changed from
to
?
How did that change affect the graph?
The general form of
the equations shown above is
![](../Resources/Assignment2/QuadraticGeneral.gif)
We can have all sorts of combinations
of a, b, & c:
![](../Resources/Assignment2/QuadExamples.gif)
The graph below shows
various values for a, where b = 1
and c = 1;
That is, a changes while b and c
are both fixed.
{Note: In each of these cases, a is a POSITIVE value}
![](../Resources/Assignment2/3.gif)
Values for a
Light Blue: 0.5; Purple: 1; Blue: 2; Red: 3
How does changing the
coefficient a seem to affect the behavior of
the graph?
What happens when a
is a Negative value?
The equations graphed
below all have negative a values.
Is that what you expected?
How does a negative
value for a change the graph?
![](../Resources/Assignment2/4.gif)
Values for a
Light Blue: -0.5; Purple: -1; Blue: -2; Red: -3
What happens when a
= 0?
Here is a graph that shows various
negative values of a that get closer and closer
to zero.
How are the graphs changing?
![](../Resources/Assignment2/aZero1.gif)
Are you sure?
What if we zoom out some?
![](../Resources/Assignment2/aZero2.gif)
How about if we zoom even further?
![](../Resources/Assignment2/aZero3.gif)
More still?
![](../Resources/Assignment2/aZero4.gif)
![](../Resources/Assignment2/aZero5.gif)
This graph shows values of a
getting closer and closer to zero from both the positive and negative
side.
![](../Resources/Assignment2/aZero6.gif)
Let's zoom back in, this time.
![](../Resources/Assignment2/aZero7.gif)
And again.
![](../Resources/Assignment2/aZero8.gif)
As a goes to zero, what can you
say about the behavior of the quadratic equation
![](../Resources/Assignment2/QuadraticGeneral.gif)
Changing
a
Can you characterize how changing
a changes the graph?
Changing
b
What happens when we
adjust the b value?
Remember, we are looking at the
general equation:
![](../Resources/Assignment2/QuadraticGeneral.gif)
The graph below shows various
positive values of b
![](../Resources/Assignment2/bVarious.gif)
Values for b
Purple: 1; Red: 2; Blue: 3; Green: 5; Light Blue - 7
What do you think will
happen when we vary negative values of b?
Can you characterize how changing
b changes the graph?
Changing
c
What affect will adjusting
c, the "constant", have?
![](../Resources/Assignment2/cVarious.gif)
Values for c
Purple: 1; Red: 2; Blue: 3; Green: 5; Light Blue - 7
What affect will negative
values of c have on a graph?
Can you characterize how changing
c changes the graph?
Ok. Take a look a the
graphs below. What can you determine about a, b,
& c?
What is changing? What causes the line with a negative slope?
![](../Resources/Assignment2/bneg.gif)
How about these? Can
you characterize a, b, & c?
![](../Resources/Assignment2/bzero.gif)