## 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:

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

And graph the points, connecting them with a smooth curve:

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

The shape of this graph is a parabola.

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:

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

The graph is shifted up

3 units from the graph of

*y* = *x*^{2}, and the vertex is

(0, 3).

Observe the graph of

*y* = *x*^{2} - 3:

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

The graph is shifted down

3 units from the graph of

*y* = *x*^{2}, and the vertex is

(0, - 3).

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

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

The graph is shifted

*left*3 units from the graph of

*y* = *x*^{2}, and the vertex is

(- 3, 0).

Observe the graph of

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

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

The graph is shifted to the

*right*3 units from the graph of

*y* = *x*^{2}, and the vertex is

(3, 0).

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