1. Polynomial Functions Quadratic Functions and Models

2. Polynomials We have looked at several functions so far. Constant Function Linear Function Squaring Function These are all examples of what we call polynomial functions.

3. Definition of a Polynomial Function Let n be a nonnegative integer and let be real numbers with. The function given by is called a polynomial function of x with degree n.

4. Degree of a Polynomial A polynomial is classified by its degree, which is the highest power of x. A constant function has degree zero. A linear function has degree one. A squaring function has degree two. This is the type of function we will be looking at now and we will call it a quadratic function.

5. Definition of a Quadratic Function Let a, b, and c be real numbers with. The function given by is called a quadratic function. We already know that the graph of this function is called a parabola.

6. Definition of a Quadratic Function Let a, b, and c be real numbers with. The function given by is called a quadratic function. We already know that the graph of this function is called a parabola. The line that splits the parabola in half is called the axis of symmetry, and this line will always intersect the vertex of the parabola.

7. Definition of a Quadratic Function Let a, b, and c be real numbers with. The function given by is called a quadratic function. We already know that the graph of this function is called a parabola. The line that splits the parabola in half is called the axis of symmetry, and this line will always intersect the vertex of the parabola. If a, the leading coefficient is positive, the parabola opens up, and has a min but no max. If a, the leading coefficient is negative, the parabola opens down, and a max but no min.

8. Graphs of Parabolas Compare the graph of and. The 1/3 makes the y values get bigger slower and makes the parabola wider than the original.

9. Graphs of Parabolas Compare the graph of and. The 2 makes the y values get bigger faster and makes the parabola thinner than the original.

10. Graphs of Parabolas Compare with the following parabolas. What changes are there? 1. 2. 3.

11. Graphs of Parabolas Compare with the following parabolas. What changes are there? 1. Same size, opens down 2. 3.

12. Graphs of Parabolas Compare with the following parabolas. What changes are there? 1. Same size, opens down 2. Thinner, opens up 3.

13. Graphs of Parabolas Compare with the following parabolas. What changes are there? 1. Same size, opens down 2. Thinner, opens up 3. Wider, opens up

14. Standard Form of a Parabola The quadratic function given by is in standard form. This is nice because it gives us the vertex and axis of symmetry. The vertex is the point (h, k) and the axis of symmetry is the vertical line x = h.

15. Example 2 Sketch the graph of and identify the vertex and axis of symmetry.

16. Example 2 Sketch the graph of and identify the vertex and axis of symmetry. We want to put the equation in what we call the standard form of a parabola. To us, this means completing the square.

17. Example 2 Complete the square: First, make sure the is a 1. Separate the constant, but keep it on the same side.

18. Example 2 Complete the square: First, make sure the is a 1. Separate the constant, but keep it on the same side. Take half of the middle term, square it, then add and subtract it from the right side of the equation.

19. Example 2 Complete the square: First, make sure the is a 1. Separate the constant, but keep it on the same side. Take half of the middle term, square it, then add and subtract it from the right side of the equation. Simplify; this in standard form.

20. Example 2 Now, using this equation, we can give the vertex and axis of symmetry. The vertex is the point (-2, -1) and the axis of symmetry is the line x = -2.

21. Standard Form You try. Put the following quadratic function in standard form and state its vertex and axis of symmetry.

22. Standard Form You try. Put the following quadratic function in standard form and state its vertex and axis of symmetry.

23. Standard Form You try. Put the following quadratic function in standard form and state its vertex and axis of symmetry.

24. Standard Form You try. Put the following quadratic function in standard form and state its vertex and axis of symmetry. Vertex ( -2, -15) axis: x = -2

25. Standard Form You try. Put the following quadratic function in standard form and state its vertex and axis of symmetry.

26. Standard Form You try. Put the following quadratic function in standard form and state its vertex and axis of symmetry. Vertex ( 3, 11) axis: x = 3

27. Finding the vertex and x-intercepts Example 3 Sketch the graph of and identify the vertex and x-intercepts.

28. Finding the vertex and x-intercepts Example 3 Sketch the graph of and identify the vertex and x-intercepts. There is another way to find the vertex without completing the square. The vertex of a parabola in this form is

29. Finding the vertex and x-intercepts Example 3 Sketch the graph of and identify the vertex and x-intercepts. Remember, in this problem a = -1, b = 6, and c = -8. So meaning that h = 3.

30. Finding the vertex and x-intercepts Example 3 Sketch the graph of and identify the vertex and x-intercepts. Remember, in this problem a = -1, b = 6, and c = -8. So meaning that h = 3. so k = 1

31. Finding the vertex and x-intercepts Example 3 Sketch the graph of and identify the vertex and x-intercepts. Remember, in this problem a = -1, b = 6, and c = -8. So meaning that h = 3. the vertex is ( 3, 1) axis is x = 3.

32. Finding the vertex and x-intercepts Example 3 Sketch the graph of and identify the vertex and x-intercepts. Remember, the x-intercepts are where the function is equal to zero. Set the equation equal to zero and solve.

33. Finding the vertex and x-intercepts Example 3 Sketch the graph of and identify the vertex and x-intercepts. Remember, the x-intercepts are where the function is equal to zero. Set the equation equal to zero and solve.

34. Finding the vertex and x-intercepts Example 3 Sketch the graph of and identify the vertex and x-intercepts. Remember, the x-intercepts are where the function is equal to zero. Set the equation equal to zero and solve.

35. Finding the vertex and x-intercepts Example 3 Sketch the graph of and identify the vertex and x-intercepts. Vertex = (3, 1) Axis: x = 3 X-intercepts (2, 0) and (4, 0)

36. Example 3 You try: Find the vertex and x-intercepts for the following. Then, sketch the graph.

37. Example 3 You try: Find the vertex and x-intercepts for the following. Then, sketch the graph. The vertex is (1, -4)

38. Example 3 You try: Find the vertex and x-intercepts for the following. Then, sketch the graph. The vertex is (1, -4) x-intercepts

39. Example 3 You try: Find the vertex and x-intercepts for the following. Then, sketch the graph. The vertex is (1, -4) x-intercepts

40. Example 3 You try: Find the vertex and x-intercepts for the following. Then, sketch the graph.

41. Example 3 You try: Find the vertex and x-intercepts for the following. Then, sketch the graph. The vertex is (-2, -2)

42. Example 3 You try: Find the vertex and x-intercepts for the following. Then, sketch the graph. The vertex is (-2, -2) x-intercepts

43. Example 3 You try: Find the vertex and x-intercepts for the following. Then, sketch the graph. The vertex is (-2, -2) x-intercepts

44. Example 4 Find the equation of a parabola. Write in standard form the equation of the parabola whose vertex is (1, 2) and that passes through the point (0, 0).

45. Example 4 Find the equation of a parabola. Write in standard form the equation of the parabola whose vertex is (1, 2) and that passes through the point (0, 0). If the vertex is (1, 2) we can start by writing:

46. Example 4 Find the equation of a parabola. Write in standard form the equation of the parabola whose vertex is (1, 2) and that passes through the point (0, 0). If the vertex is (1, 2) we can start by writing: If (0, 0) is on the parabola, we substitute those values for x and y and solve for a.

47. Homework Pages 270-271 1-8 all 17-20 all 37,39, 43, 45