An easy method for
graphing parabolas involves preparing a chart.
Of course, the graphing calculator can also be used for graphing.
Example 1:
Graph the parabola y
= x^{2}  4x on the interval [1,5].
(Remember: [1,5] means from x = 1 to x = 5
inclusive.)
Prepare a chart:
x 
x^{2}
 4x 
y 
1 
(1)^{2}
 4(1) 
5 
0 
(0)^{2}
 4(0) 
0 
1 
(1)^{2}
 4(1) 
3 
2 
(2)^{2}
 4(2) 
4 
3 
(3)^{2}
 4(3) 
3 
4 
(4)^{2}
 4(4) 
0 
5 
(5)^{2}
 4(5) 
5 

Plot the
points generated in the chart. Draw a
smooth curve through the points.
The points where the graph
crosses the
xaxis are called the roots
of
0 = x^{2}  4x.
This parabola crosses the xaxis
at (0,0) and (4,0).


In this example,
the interval used for preparing the chart was given in the question.
Consequently, the turning point of the parabola fell within the
interval. If the question had NOT told us the interval, how
would we have known which values to place in the chart to ensure that we would
see the turning point of the parabola? To guarantee that the
points you choose for your chart will show the turning point, start by determining the
axis of symmetry of the parabola.
The axis
of symmetry is a vertical line passing through the turning point
of a parabola.
In our first example,
the turning point was (2,4).
The equation of the axis of symmetry is the equation of the vertical line
passing through (2,4), or in this case x = 2.
Parabolas are of
the quadratic form: y = ax^{2} +
bx + c 


If
a
is positive, the parabola
opens upward and has a minimum point.
The axis of symmetry is
x = (b)/2a 
If
a
is negative, the parabola opens downward and has a maximum point.
The axis of symmetry is
x = (b)/2a. 
Example 2:
Graph the parabola y
= x^{2} + 6x  1
(no interval
specified)
Rather than picking numbers at
random to form our chart of values, let's first find the axis of
symmetry. This will guarantee that our chart will graph the
turning point of the parabola and give us a good graph.
To
find
the axis of symmetry,
use the formula
x =
b/2a
In this example, a = 1 and b = 6.
Substituting gives: x = (6)/2(1) = 6/2 = 3
Axis
of symmetry: x = 3
Since the
xcoordinate of the turning point is 3, use this
value as the middle value for x
in the chart. Include 3 values above and below 3
in the chart.
Substitute
each value of x into the quadratic equation (the
parabola), to find the corresponding values for y and
complete the table. When the table is complete, plot the
points and draw the graph.

Set up the table.

x 
y 
6
5
4
3
2
1
0 
1
6
9
10
9
6
1 
Complete the table.





See how to use your
TI83+/84+ graphing calculator
when working with parabolas.
Click calculator. 

