 # Essential Question: How do you determine whether a quadratic function has a maximum or minimum and how do you find it?

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Essential Question: How do you determine whether a quadratic function has a maximum or minimum and how do you find it?

 All quadratic functions are parabolas:  “U” shaped  Standard form: y = ax 2 + bx + c If a is positive, the graph opens up If a is negative, the graph opens down  Always have a vertex that is either the maximum or minimum If the parabola opens up, it has a minimum If the parabola opens down, it has a maximum  Always have one y-intercept, which is at (0, c)

 Determine the y-intercept and whether the graph opens up or opens down  y = x 2 + 8x – 1 Opens: y-intercept:  y = - 1 / 2 x 2 + 2 Opens: y-intercept:  y = 2x 2 – x Opens: y-intercept: (0, -1) (0, 2) (0, 0) Up Down

 In standard form, the axis of symmetry is found by using the equation:  Take the number in front of the “x” (that’s “b”) and the number in front of the “x 2 ” (that’s “a”), and plug them into the equation above.

 Example:  For the equation: y = x 2 – 2x – 3 find the axis of symmetry So x=  The axis of symmetry also is the first step in finding the vertex of a parabola a = 1, b = -2, c = -3

 The vertex of a parabola is located at:  This means that: The x-coordinate is at –b / 2a The y-coordinate is the number you get when you plug the x-value back into the original function.  Example  In the graph: y = x 2 – 2x – 3, find the vertex. We know x = 1 (last slide), so substitute 1 in for x and solve. y = (1) 2 – 2(1) – 3 = -4 The vertex is at (1, -4)

 Find the axis of symmetry and the vertex  y = -x 2 + 4x + 3 Axis of symmetry: Vertex:  y = - 1 / 3 x 2 – 2x – 4 Axis of symmetry: Vertex: x = 2 x = -3 (2, 7) (-3, -1)

 Assignment  Page 248  Problems 1-21, odd I GNORE THE DIRECTIONS !!! Tell me: a) Whether the graph opens up or opens down b) The y-intercept c) The axis of symmetry d) The vertex

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