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The General Quadratic Function Students will be able to graph functions defined by the general quadratic equation.

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FHSQuadratic Function2 The General Quadratic Function The general quadratic function may be written as: f(x) = ax 2 + bx + c or y = ax 2 + bx + c, where a, b, and c are real numbers and a ≠ 0. Why can’t a be equal to 0 ? Because if a were equal to 0, then there would be no x 2 term. Then we would not have a quadratic equation.

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FHSQuadratic Function3 Vertex of the Parabola When we have a quadratic equation written as y = ax 2 + bx + c, the vertex of the parabola is the lowest or highest point on the graph. The x - coordinate of the vertex is. The y -coordinate of the vertex can be found by substituting the x -value for x in the original equation and finding y.

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FHSQuadratic Function4 Axis of Symmetry The axis of symmetry for the parabola is the vertical line that passes through the vertex. The equation of that line is

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FHSQuadratic Function5 The y -intercept Another point that helps in graphing the parabola is the y -intercept. To find the y -intercept, substitute 0 into the equation in place of x. The value that you get for y is the y - intercept. You can then use the graph and reflect the y-intercept across the axis of symmetry to find another point on the graph.

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FHSQuadratic Function6 The Vertex Form To find the vertex form of the quadratic equation, find the vertex and substitute it into the following form: where (h, k) is the vertex of the parabola.

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FHSQuadratic Function7 Example Given the equation find each of the following: 1.The coordinates of the vertex 2.The axis of symmetry 3.The y-intercept 4.The reflection of the y-intercept 5.The vertex form of the equation 6.The graph of the equation

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FHSQuadratic Function8 Example: Vertex Find the coordinates of the vertex for the graph of. Find the x -coordinate for the vertex. Find the y -coordinate for the vertex. Thus, the vertex is (1, 3)

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FHSQuadratic Function9 Example: Axis of Symmetry and y -intercept For this equation: The axis of symmetry is: The y-intercept is:

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FHSQuadratic Function10 Example: Vertex Form Since the vertex for this equation: was (1, 3) and a = 2, we substitute those into the following: with vertex (h, k) to get the following vertex form:

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FHSQuadratic Function11 Example When we graph this quadratic equation, the parabola opens up. The vertex is (1, 3). The axis of symmetry is x = 1. The y-intercept is 5. The reflected point is (2, 5). So the graph is: (1, 3) axis of symmetry

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