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Quadratics

Graphing Parabolas

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Graphing Parabolas, page 2

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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

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