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Graphing Parabolas Graphing y = x2 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: Data Table for y = x2 Graph of y = x2 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 = 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: Graph of y = x2 + 3 Observe the graph of y = x2 - 3: Graph of y = x2 - 3 We can also shift the vertex left and right. Observe the graph of y = (x + 3)2: Graph of y = (x + 3)2 Observe the graph of y = (x - 3)2: Graph of y = (x - 3)2 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 = 2x2: Data Table for y = 2x2 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: Graph of y = 2x2 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 |
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