Parabolas are written in the
form y=ax^{2} +bx +c.
(Sometimes they are written in vertex form which is y= a(x-h)^{2}
+h, but we are going to leave that for later.) We will vary the coefficients a, b, and c to see what affect
the changes have on the shape and location of the parabola.

**Step 1: Let a=1, b=0** and
**c=0**. Our graph becomes y = x^{2 }. This graph of
the parabola is called the parent graph.
All other graphs are variations of this one. Where is the vertex of the parabola?

Vertex
(0,0)

**Step 2: **Vary the values for a, but let b=0 and c=0. We will keep the green graph (our parent
graph) so that we can use it for comparison purposes. What happens to the parabola as the value of **a **increases?
What happens when the value of **a** is a negative number?
Does the vertex stay in the same position? Try different values for **a** in graphing calculator.

As you can see from the
graphs, when the value of **a > 0 **the
parabola opens up. When the value
of **a<0**, the parabola opens
down. Yes, the vertex remains at the point (0,0). As **a** increases, the parabola becomes thinner. The parabola
widens as **a** approaches 0. What graph would we have if a=0? (y= 0x^{2}+bx +c)

By substituting zero for **a** we now have the equation, y=0x^{2}+bx
+c. This simplifies to y= bx
+c. This is familiar to us as the
equation of a line.

** **

**Step 3: **y= ax^{2} +bx +c Let **a=1**, **b=0** and **vary c.**

Use graphing calculator to
test values for c. See the
examples below after you have tried some on your own.

Examples: y = x^{2 } V: (0,0) ; y =
x^{2 }+ 3 V:( 0, 3) ; y
= x^{2} +5 V: (0,5)

y = x^{2} + (-3) V: (0, -3); y = x^{2} –5 V:(0, -5)

The value of **c **causes the parabola to shift up or down. The shape of the parabola has not
changed. It has just moved up or
down according to the value of **c**.
Remember all this time **a=1** and **b=0**. The y value of the vertex moves up or down **c** units.
When **c** is positive, the
parabola shifts up. When **c** is negative, the parabola shifts down.

**Step 4**: y = ax^{2} +bx +c Let a=1, c=2, and vary b. y = x^{2} + bx +2

Look at the three graphs
below. Two graphs have shifted up
(c=2) from the position of the parent graph. However, they have not shifted up 2 units. What effect does **b **(the coefficient of the **x** term) have on the movement of the parabola? Try several graphs in graphing
calculator to see if you can determine any relationships.

y= x^{2}
V: (0,0) y = x^{2}
+1x + 2 V: y = x^{2}
+ 2x +2 V: (-1, 1)

Did you determine that even
though **b** was positive, the graph
moved to the left? Is there a
relationship between **b** and **a**? Do we
notice any relationship changes in the **x** value of the vertex?

Below are two additional
graphs where b is negative.

y= x^{2} – 2x
+2 V: (2,-2)
y= x^{2} –4x +2 V: (1, 1)

If **c** is positive why is does the graph move down? The value of y is dependent on the
value of **x**. The value of the **x **coordinate is equal to . Look back at
the values of **a** and **b** and compare them to the x value of the vertex. Do you see the relationship that
exists? Once we know the value of **x**, we can put it in the quadratic equation to find the
value of **y**. It is important to pay attention to all
parts of the equation.

Click HERE
to see a graph of a parabola for values of b between –5 and 5.

To summarize, the value of **a
**affects the width of the parabola and
the direction in which it opens.
The value of **c **determines
whether the parabola will shift up or down. Lastly, the value of **b **determines whether the graph will shift to the left or
right.