Warm-Up: December 15, 2011  Divide and express the result in standard form.

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Presentation transcript:

Warm-Up: December 15, 2011  Divide and express the result in standard form

Homework Questions?

Quadratic Functions Section 2.2

Quadratic Functions  A quadratic function is any function that can be written in the form  The graph of a quadratic function is a parabola.  Every parabola has a vertex at either its minimum or its maximum.  Every parabola has a vertical axis of symmetry that intersects the vertex.

Example Graphs Vertex Axis of Symmetry

Standard Form of a Quadratic Function  Vertex is at (h, k)  Axis of symmetry is the line x=h  If a>0, the parabola opens upward, U  If a<0, the parabola opens downward,

Graphing Quadratics in Standard Form 1. Determine the vertex, (h, k) 2. Find any x-intercepts by replacing f(x) with 0 and solving for x 3. Find the y-intercept by replacing x with 0 4. Plot the vertex, axis of symmetry, and y-intercepts and connect the points. Draw a dashed vertical line for the axis of symmetry. 5. Check the sign of “a” to make sure your graph opens in the right direction.

Example 1  Graph the quadratic function.  Give the equation of the parabola’s axis of symmetry.  Determine the graph’s domain and range.

You-Try #1  Graph the quadratic function.  Give the equation of the parabola’s axis of symmetry.  Determine the graph’s domain and range

Graphing Quadratics in General Form  General form is  The vertex is at  x-intercepts can be found by quadratic formula (or sometimes by factoring and zero product property)  y-intercept is at (0, c)  Graph the parabola using these points just as we did before.

Example 3  Graph the quadratic function.  Give the equation of the parabola’s axis of symmetry.  Determine the graph’s domain and range

You-Try #3  Graph the quadratic function.  Give the equation of the parabola’s axis of symmetry.  Determine the graph’s domain and range

Minimum and Maximum  Consider  If a>0, then f has a minimum  If a<0, then f has a maximum  The maximum or minimum occurs at  The maximum or minimum value is

Example 4 (Page 266 #44)  A football is thrown by a quarterback to a receiver 40 yards away. The quadratic function models the football’s height above the ground, s(t), in feet, when it is t yards from the quarterback. How many yards from the quarterback does the football reach its greatest height? What is that height?

You-Try #4 (Page 266 #43)  Fireworks are launched into the air. The quadratic function models the fireworks’ height, s(t), in feet, t seconds after they are launched. When should the fireworks explode so that they go off at the greatest height? What is that height?

Assignment  Page 264 #1-8 ALL (use your graphing calculator for 5-8), #9-41 Every Other Odd