Consider the general equation of a parabola:

We know that the graph of a quadratic will
give us a parabola. How will changing **a**,**b**, and **c** affect
the vertex of the parabola?

First let's look at the standard form for the equation of a parabola:

The vertex in the above form is at **(h,k)**. We can put the general equation
in terms of **a**,**b**, and **c** in the standard form by completing
the square:

As we can see **b** and
**a** will affect the x coordinate of the vertex, while **a**,**b**,and
**c** all affect the y coordinate.

Now let's look at some graphs:
First we will let **b**=0,**c**=0. From the solutions for **h** and **k**
we expect the vertex to be at (0,0).

We know that with **b** and **c** equal to zero our vertex will
be at (0,0), so what affect will changing **a** have on the parabola?
Lets change **a** and look at the graphs:

The purple graph is **a**=.25,the
red **a**=.5,the
blue **a**=.75,the
green **a**=1
and the aqua **a**=1.5.
So we see as **a** increases the parabola gets thinner. If we look at
the negatives for the previous values of **a** we get:

So negative values of **a** cause the parabolas to open downwards.
Remember the coordinates of the vertex in terms of **a**,**b**,**c**
where are:

So let **a**=1 and **b** = 0. As we change **c** we expect the
y coordinate of the vertex to change. As **c** increases the vertex will
get higher. Lets look at a few parabolas:

The purple parabola was obtained when **c **= 0, the red when **c **= 1 and the blue when **c
**= - 1.

Next we will look at the cases when **a**=1, **c**=0 and **b**
varies. The coordinates of the vertex will be: when we
plot these as x,y coordinates with b varying we get:

This is the locus of vertices for **a**=1 and **c**=0 with **b**
varying. If we look at several parabolas on top of the locus of vertices
we see:

**red:b = 0,blue:b
= 1,green:b = -1,aqua:b
= 2,yellow:b=-2,lavender:b
= 3,and,gray:b = -3:**

Now if we keep **a**=1 and vary **c** and **b** the coordinates
of our vertex would be:

This will shift our parabola up by the constant c. If we look at the
previous graph with **c**= 1, instead of zero, we get:

Our locus of vertices is now not correct. If we move it up 1 unit will it be correct?

Now for for **c** = -1 we would expect to shift the family of parabolas
with **b** varying and **a** = 1 down one:

Now what happens when we vary **a**? Lets look at the values of the
coordinates of the vertex in terms of the coefficients of our quadratic:

.

As we can see changing **a** will affect both the x and y-coordinates
of the vertex. Lets look at the family of parabolas with **b** varying
and **c**=0 with different values of **a**. First with **a**=2
we will get the following:

We can compare this to the graphs when **a**=1,**c**=0:

The equation of the vertex in the first graph with **a**=2,**c**=0
is and for the second graph with **a**=1,**c**=0
the equation of the vertex is . So we see that the
parabola with **a**=2,**c**=0 has a vertex with coordinates that are
one-half the coordinates of the vertex when **a**=1,**c**=0.

We looked at the graphs of as
we changed **a**,**b**,**c**. We saw changing **a** with **b**,**c**,constant we changed the
width of our parabola. As we let **a** go to zero
our parabola increased in width. For a positive **a** the
parabola opened upwards
and and a negative **a** caused our parabola to open downwards. We saw that changing **c** caused a vertical shift upwards
of the vertex by the value c and downwards the value c for negative **c.**
When we changed **b**
with **a** and **c** constant we found that the locus of vertices of our parabola was a parabola
itself.

We saw the locus of vertices of our parabola given by, ,

with **a** positive and **b** varying
was a parabola that opened downwards. What affect would a negative **a**
have on the locus of vertices parabola?

Recall the standard form of a parabola:. When we put our equation ,, in standard form we got .

We see that . If y =
k-p is the equation of the directrix of the standard parabola and the focus
is at (h,k+p) how does **a,b,c** affect the directrix and focus?

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