Quadratics
Graphing Parabolas
Graphing y = x 2
We have already discovered how to graph linear functions. But what does the graph of
y = x
2
look like? To find the answer, make a data table:
Note that the parabola does not have a constant slope. In fact, as x increases by 1 , starting with x = 0 , y increases by 1, 3, 5, 7,… . As x decreases by 1 , starting with x = 0 , y again increases by 1, 3, 5, 7,… .
Graphing y = (x - h)2 + k
In the graph of y = x 2 , the point (0, 0) is called the vertex. The vertex is the minimum point in a parabola that opens upward. In a parabola that opens downward, the vertex is the maximum point.
We can graph a parabola with a different vertex. Observe the graph of
y = x
2 + 3
:
Observe the graph of y = x 2 - 3 :
We can also shift the vertex left and right. Observe the graph of
y = (x + 3)2
:
%5E2.gif)
Observe the graph of y = (x - 3)2 :
%5E2.gif)
In general, the vertex of the graph of
y = (x - h)2 + k
is
(h, k)
. For example, the vertex of
y = (x - 2)2 + 1
is
(2, 1)
:
%5E2+1.gif)
The axis of symmetry is the line which divides the parabola into two symmetrical halves. It is given by the equation
x = h
. For example, the line of symmetry in the graph of
y = (x - 2)2 + 1
is
x = 2
:
Graphing y = a(x - h)2 + k
In addition to shifting the parabola up, down, left, and right, we can stretch or shrink the parabola vertically by a constant. We can make a data table for the graph of
y = 2x
2
:
In general, in the graph of y = a(x - h) + k , as x increases or decreases by units of 1 starting from the vertex, y increases by 1a, 3a, 5a, 7a,… .
Here is the graph of
y = 2x
2
:
Graphing y = - a(x - h)2 + k
Sometimes, the coefficient in front of (x - h) is negative. If this is the case, the parabola opens downward. In the graph of y = - a(x - h) + k , the vertex and axis of symmetry are still (h, k) and x = k , but as x increases or decreases by units of 1 starting from the vertex, y decreases by 1a, 3a, 5a, 7a,… .
For example, here are the data table and graph for
y = - (x - 2)2 + 3
:
%5E2+3.gif)
