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

2. 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.

3. 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.

4. 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

5. 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.

6. 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.

7. 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

8. 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)

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

10. 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:

11. 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