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:

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 left 3 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 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) :

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 :

Graphing y = a(x - h)² + k

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 = 2x ² :

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 = 2x ² :

Graphing y = - a(x - h)² + k

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 :