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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

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 :

Axis of Symmetry

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 :

Data Table for 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 :

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 :

Graph of y = - (x - 2)2 + 3