C. Quadratic Functions

1. C. Quadratic Functions Math 20 Pre-Calculus P20.7 Demonstrate understanding of quadratic functions of the form   y=ax²+bx+c and of their graphs, including: vertex, domain and range, direction of, opening, axis of symmetry, x- and y-intercepts.

2. Key Terms:

3. Quadratic Functions occur in a wide variety of real world situations. In this unit we will investigate functions and use them in math modelling and problem solving.

4. 1. Vertex Form • P20.7 • Demonstrate understanding of quadratic functions of the form   y=ax²+bx+c and of their graphs, including: • vertex • domain and range • direction of opening • axis of symmetry • x- and y-intercepts.

5. 1. Vertex Form • Investigate p. 143

6. The graph of a Quadratic Function is a parabola • When the graph opens up the vertex is the lowest point and when it opens down the vertex is the highest point

7. The y-coordinate of the vertex is called the min value or max value depending of which way it opens. • The parabola is symmetrical about a line called the axis of symmetry. The line divides the graph into two equal halves, left and right. • So if you know the a of s and a point you can find another point (unless the point is the vertex)

8. The A of S intersects the vertex • The x-coordinate of the vertex is the equation of the A of S.

10. Quadratic Function in vertex formf(x) = a(x-p)2+q are very easy to graph. • a, p, and q tell you what you need. • (p,q) = Vertex • Opens up +a • Opens down –a • Larger a = narrower parabola • Smaller a = wider parabola

11. Example 1

12. Example 2

13. Example 3

14. Example 4

15. Key Ideas p.156

17. Practice • Ex. 3.1 (p.157) #1-14 #4-18

18. 2. Standard Form • P20.7 • Demonstrate understanding of quadratic functions of the form   y=ax²+bx+c and of their graphs, including: • vertex • domain and range • direction of opening • axis of symmetry • x- and y-intercepts.

19. 2. Standard Form

20. Recall that the Standard form of a quadratic function is f(x) = ax2+bx+c or y = ax2+bx+c • Where a, b, c are real numbers and a ≠ 0 • a determines width of graph (smaller a = wider graph) and opening (+a up and –a down) • b shifts the graphs left and right • c shifts the graph up and down

21. We can expand f(x) = a(x-p)2+q to get f(x) = ax2+bx+c , which will allow us to see the relation between the variable coefficients in each.

22. So, • b = -2ap or And • c = ap2 + q or q = c – ap2

23. Recall that to determine the x-coordinate of the vertex, you use x = p. • So the x-coordinate of the vertex is

24. Example 1

25. Example 2

26. Example 3

27. Key Ideas p.173

30. Practice • Ex. 3.2 (p.174) #1-9, 11-17 odds #5-25 odds

31. 3. Completing the Square • P20.7 • Demonstrate understanding of quadratic functions of the form   y=ax²+bx+c and of their graphs, including: • vertex • domain and range • direction of opening • axis of symmetry • x- and y-intercepts.

32. 3. Completing the Square • You can express a quadratic function in vertex form, f(x) = a(x-p)2+q or standard form f(x) = ax2+bx+c • We already know we can go from vertex to standard by just expanding • However to graph by hand it is much easier if the function is in vertex form because we have the vertex, axis of symmetry and max or min of the graph

33. So to be able to turn a standard form function into vertex form would be advantageous. • This process is called Completing the Square

34. What we want to be able to do is rewrite the trinomial as a binomial squared. (x+5)(x+5) = (x+5)2

35. Lets complete the square:

36. If there is a coefficient in front of the x2 term we have to add a couple steps. Complete the Square:

37. Example 1

38. Example 2

39. Example 3

40. Example 4

41. Key Ideas p.192

42. Practice • Ex. 3.3 (p.192) #1-9, 10-18 evens #1-9 odds in each, 10-28 evens