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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