Section 2.6 Quadratic functions

2. Section 2.6   Quadratic functions

3. A baseball is “popped” straight up by a batter. The height of the ball above the ground is given by the function: y = f(t) = −16t2 + 64t + 3, where t is time in seconds after the ball leaves the bat and y is in feet. Let's use our calculator: Page 88

4. Let's use our calculator: Y= → \Y1= −16x2+64x+3 Page N/A

5. Let's use our calculator: Y= → \Y1= −16x2+64x+3 Window → Graph Page N/A

6. Although the path of the ball is straight up and down, the graph of its height as a function of time is concave down. Page 88

7. The ball goes up fast at first and then more slowly because of gravity. Page 88

8. The baseball height function is an example of a quadratic function, whose general form is y = ax2 + bx + c. Page 89

9. Finding the Zeros of a Quadratic Function Page 89

10. Finding the Zeros of a Quadratic Function Back to our baseball example, precisely when does the ball hit the ground? Page 89

11. Finding the Zeros of a Quadratic Function Back to our baseball example, precisely when does the ball hit the ground? Or: For what value of t does f(t) = 0? Page 89

12. Finding the Zeros of a Quadratic Function Back to our baseball example, precisely when does the ball hit the ground? Or: For what value of t does f(t) = 0? Input values of t which make the output f(t) = 0 are called zeros of f. Page 89

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18. Let's use our calculator: Y= → \Y1= −16x2+64x+3 Window → Graph Page N/A

19. Now let's use the TI to find the zeros of this quadratic function: Page N/A

20. 2nd Trace 2: zero Left Bound ? Right Bound? Guess? Page N/A

21. zero X=4.0463382 Y=-1E-11 Page N/A

22. Example #1: Find the zeros of f(x) = x2 − x − 6. Page 89

23. Example #1: Find the zeros of f(x) = x2 − x − 6. Set f(x) = 0 and solve by factoring: x2 − x − 6 = 0 (x-3)(x+2) = 0 x = 3 & x = -2 Page 89

24. Example #1: Find the zeros of f(x) = x2 − x − 6. Let's use our calculator: Page 89 Example #1

25. Let's use our calculator: Y= → \Y1= x2-x-6 Page N/A

26. Let's use our calculator: Y= → \Y1= x2-x-6 Zoom 6 gives: Graph Page N/A

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28. Now let's use the TI to find the zeros of this quadratic function: Page N/A

29. 2nd Trace 2: zero Left Bound ? Right Bound? Guess? Page N/A

30. zero x=-2 y=0 Page N/A

31. 2nd Trace 2: zero Left Bound ? Right Bound? Guess? Page N/A

32. zero x=3 y=0 Page N/A

33. Example #3 Figure 2.29 shows a graph of: What happens if we try to use algebra to find its zeros? Page 89 Example #3

34. Let's try to solve: Page 89

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36. Conclusion? Page 90

37. Conclusion? There are no real solutions, so h has no real zeros. Look at the graph again... Page 90

38. What conclusion can we draw about zeros and the graph below? y x Page 89

39. h has no real zeros. This corresponds to the fact that the graph of hdoes not cross the x-axis. y x Page 89

40. Let's use our calculator: Y= → \Y1= (-1/2)x2-2 Page N/A

41. Let's use our calculator: Y= → \Y1= (-1/2)x2-2 Window Graph Page N/A

42. y x Page N/A

43. 2nd Trace 2: zero Left Bound ? Right Bound? Guess? Page N/A

44. 2nd Trace 2: zero Left Bound ? Right Bound? Guess? ERR:NO SIGN CHNG 1:Quit Page N/A

45. Concavity and Quadratic Functions Page 90

46. Concavity and Quadratic Functions Unlike a linear function, whose graph is a straight line, a quadratic function has a graph which is either concave up or concave down. Page 90

47. Example #4 Let f(x) = x2. Find the average rate of change of f over the intervals of length 2 between x = −4 and x = 4. What do these rates tell you about the concavity of the graph of f ? Page 90 Example #4

48. Let f(x) = x2 Between x = -4 & x = -2: Page 90

49. Let f(x) = x2 Between x = -2 & x = 0: Page 90

50. Let f(x) = x2 Between x = 0 & x = 2: Page 90