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:
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,….

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:
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:
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:
The graph is shifted left 3 units from the graph of y = x², and the vertex is (- 3, 0).

Observe the graph of y = (x - 3)²:
The graph is shifted to the right 3 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):

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:

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 = 2x²:
Here, the y increases from the vertex by 2, 6, 10, 14,…; that is, by 2(1), 2(3), 2(5), 2(7),….

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 1a, 3a, 5a, 7a,….

Here is the graph of y = 2x²:

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 1a, 3a, 5a, 7a,….

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