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:
And graph the points, connecting them with a smooth curve:
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:
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:
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}:
The graph is shifted left3 units from the graph of y = x^{2}, and the vertex is (- 3, 0).
Observe the graph of y = (x - 3)^{2}:
The graph is shifted to the right3 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):