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Graphing y = x2

We have already discovered how to graph linear functions. But what does the graph of y = x2 look like? To find the answer, make a data table:

Data Table for y = x2
And graph the points, connecting them with a smooth curve:
Graph of y = x2
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 = x2, 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 = x2 + 3:

Graph of y = x2 + 3
The graph is shifted up 3 units from the graph of y = x2, and the vertex is (0, 3).

Observe the graph of y = x2 - 3:
Graph of y = x2 - 3
The graph is shifted down 3 units from the graph of y = x2, 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 left3 units from the graph of y = x2, 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 right3 units from the graph of y = x2, 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

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