1. Write each expression in standard polynomial form. Welcome! Pick up a new notes, then complete the problems below. 1.2. 3.4. 5.

2. SECTION 2.2 QUADRATIC FUNCTIONS

3. Graphs of Quadratic Functions

4. Quadratic Functions  Quadratic functions of the standard form f(x) = ax 2 + bx + c  The graph of a quadratic function is called a parabola. Parabola’s are “ U ” shaped.  Parabolas have reflection symmetry with respect to the line called the axis of symmetry. If folded over this line the two halves match exactly.

5. Graphs of Quadratic Functions Parabolas Minimum Vertex Axis of symmetry Maximum

6. Graphing Quadratic Functions in Vertex Form

7. Vertex Form of a Quadratic  f (x) = a ( x – h)2 h)2 + k, a ≠ 0, where ( h, k ) is the vertex of the parabola.  The line of symmetry is x = h.h.  If a > 0, then the parabola opens up. If a < 0, then the parabola opens down.  Find x -intercepts by solving f (x) = 0.0.  Find the y -intercepts by solving f (0).  Plot the intercepts, the vertex, and any additional points necessary. Connect these points with a smooth “U” shaped curve.

8. f ( x ) = ( x - 5) 2 + 3 Vertex: (0, 0) f ( x ) = x 2 Vertex: (5, 3) Axis of symmetry x = 5

9. f ( x ) = - ( x + 8) 2 - 4 f ( x ) = - ( x + 8) 2 - 4 Vertex: (0, 0) f ( x ) = - x 2 Vertex: (-8, -4) Axis of symmetry x = -8 Axis of symmetry x = -8

11. Find the vertex.

12. Using Vertex Form                     x y

13. Using Vertex Form                     x y

14. Try these!  Solve the equation by factoring 1. 2. 3. 4.

15. Graph the quadratic function f ( x ) = -( x +2) 2 + 4 and state the vertex, axis of symmetry, x - and y -intercepts. Then convert to standard form.

16. Graph the quadratic function f ( x ) = ( x – 3) 2 – 4 Page 298 Problems 17-26

17. Graphing Quadratics in the Standard form f ( x ) = ax 2 + bx + c

18. Standard Form We identify the vertex of a parabola whose equation is in the standard form by completing the square. See next slide for finished equation

19. Ta Da!! Vertex Form From Standard Form

20. Blast from the Past…  Submit the Thanksgiving worksheets that you were given on Wednesday and had to complete over the break. If you were out, you were excused from this work.  Turn to page 339 to complete problems 8 and 9, then turn to page 382 to complete problems 13 and 14.

21. Finding the Vertex from Standard Form  When the equation is f (x) = ax 2 + bx + c, the vertex is…  You would then proceed to graph the equation the same way as before.  Determine if it opens up or down.  Plot the vertex.  Find the x-intercepts and y-intercepts and plot them.  Connect the points with your smooth curve.

22. Using the form f(x) = ax 2 + bx + c                     x y

23. Find the vertex, axis of symmetry, x-intercepts and y-intercept, open up/down of the function f ( x ) = - x 2 - 3 x + 7

24. Graph the function f ( x ) = - x 2 - 3 x + 7. Use the graph to identify the domain and range. Page 298 Problems 27-38

25. Graphing Calculators are needed!  Go get your grapher, and log in. Remember you should have a paper in your NB with your log in and password on it!  then complete problems 45-48 on page 298

26. Minimum and Maximum Values of Quadratic Functions

28. For the function f ( x ) = - 3 x 2 + 2 x – 5 Without graphing determine if it has a min or max, then find it. Identify the function’s domain and range. Page 298 Problems 39-44

29. Type the equation into f ( x ) = − 3 x 2 + 2 x – 5 Graphing Calc – Min/Max Hit b and select option 6: Analyze Graph, then choose option 3: Maximum because our graph opens down. Move the finger to the left (lower bound) of the vertex and click (or press enter). Repeat for the upper bound, but on the right side of the vertex. What happens?

30. Applications of Quadratic Functions

32. You have 64 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area? Page 299-300 Problems 57-76

33. You can always use Graphing Calculator Problem continued on the next slide

34. Graphing Calculator- continued

35. What is your equation? MKwh 18.79 28.16 36.25 45.39 56.78 68.61 78.78 810.96 Quadratic Regression on TI-Nspire Put the data from the table into the table on your calculator. When you are done, hit /~ to add 5: Data & Statistics page. Click to add your labels on the x - and y -axes. Now select b, 4: Analyze, then 6:Regression. Make sure you select the appropriate regression for the shape of our data.

36. (a) (b) (c) (d)

37. (a) (b) (c) (d) More practice can be found on page 298-300 Prob. 45-56 (b)