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:

Data Table for
*y* = *x*
^{2}

Graph of
*y* = *x*
^{2}

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,…
.

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
:

Graph of
*y* = *x*
^{2} + 3

Observe the graph of

Graph of
*y* = *x*
^{2} - 3

We can also shift the vertex left and right. Observe the graph of
*y* = (*x* + 3)^{2}
:

Graph of
*y* = (*x* + 3)^{2}

Observe the graph of

Graph of
*y* = (*x* - 3)^{2}

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)
:

Graph of
*y* = (*x* - 2)^{2} + 1

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
:

Axis of Symmetry

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* = 2*x*
^{2}
:

Data Table for
*y* = 2*x*
^{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
1*a*, 3*a*, 5*a*, 7*a*,…
.

Here is the graph of
*y* = 2*x*
^{2}
:

Graph of
*y* = 2*x*
^{2}

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
1*a*, 3*a*, 5*a*, 7*a*,…
.

For example, here are the data table and graph for
*y* = - (*x* - 2)^{2} + 3
:

Graph of
*y* = - (*x* - 2)^{2} + 3