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Note Every parabola has an axis of symmetry. In Example 1, the y-axis is the axis of symmetry, and in Example 2, the axis of symmetry is the vertical line x 2. If you were to fold each graph along its axis of symmetry, one half of the graph would lie on top of the other. Each side is a mirror image of the other. 2 Graph a Quadratic Equation of the Form y ax2 bx c How do we graph a more general parabola, one in the form y ax2 bx c? First, we must fi nd the vertex. An important property of a parabola is that if a parabola has two x-intercepts, then the x-coordinate of the vertex is always halfway between the x-coordinates of the x-intercepts of the parabola. This will always be true and will help us derive a formula to fi nd the vertex of a parabola with an equation like y x2 4x 3. Recall from previous chapters that to fi nd the x-intercepts of the graph of an equation, we let y 0 and solve for x. Therefore, to fi nd the x-intercepts of the graph of y ax2 bx c, let y 0 and solve for x. 0 ax2 bx c We can use the quadratic formula to solve for x, giving us the following x-coordinates of the x-intercepts: x b 2b2 4ac 2a   and  x b 2b2 4ac 2a Since the x-coordinate of the vertex of a parabola is halfway between the x-intercepts, we fi nd the average of the x-coordinates of the x-intercepts to fi nd the x-coordinate of the vertex. x-coordinate of the vertex 1 2 ab 2b2 4ac 2a b 2b2 4ac 2a b 1 2 ab 2b2 4ac b 2b2 4ac 2a b 1 2 a2b 2a b b 2a Procedure Finding the Vertex of a Parabola of the Form y ax2 bx c (a 0) The x-coordinate of the vertex of a parabola written in the form y ax2 bx c (a 0) is x b 2a . To fi nd the y-coordinate of the vertex, substitute the x-value into the equation and solve for y. www.mhhe.com/messersmith SECTION 10.4 Graphs of Quadratic Equations 633


messersmith_power_introductory_algebra_1e_ch4_7_10
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