Sections 4.3 and 4.4 Quadratic Functions and Their Properties

1. Sections 4.3 and 4.4 Quadratic Functions and Their Properties

2. Quadratic Function A quadratic function of the variable x is a function that can be written in the form where a, b, and c are fixed real numbers. Also, the domain of f is all real numbers, namely, (, ). Example:

3. Quadratic Function The graph of a quadratic function is a parabola. a < 0 a > 0

4. Vertex, Intercepts, Symmetry Vertex coordinates are: y – intercept is: symmetry x – intercepts are solutions of

5. Graph of a Quadratic Function Example 1:Sketch the graph of Vertex: y – intercept x – intercepts

6. Graph of a Quadratic Function Example 1:Sketch the graph of Absolute Minimum of f is: Absolute Maximum of f is: Range of f is:

7. Graph of a Quadratic Function Example 2:Sketch the graph of Vertex: y – intercept x – intercepts

8. Graph of a Quadratic Function Example 2:Sketch the graph of Absolute Minimum of f is: Absolute Maximum of f is: Range of f is:

9. Graph of a Quadratic Function Example 3:Sketch the graph of Vertex: y – intercept x – intercepts no solutions

10. Graph of a Quadratic Function Example 3:Sketch the graph of Absolute Minimum of f is: Absolute Maximum of f is: Range of f is:

11. Applications Example: For the demand equation below, express the total revenue R as a function of the price p per item and determine the price that maximizes total revenue. Maximum is at the vertex, p = $100

12. Applications Example: As the operator of Workout Fever Health Club, you calculate your demand equation to be q0.06p + 84 where q is the number of members in the club and p is the annual membership fee you charge. 1. Your annual operating costs are a fixed cost of $20,000 per year plus a variable cost of $20 per member. Find the annual revenue and profit as functions of the membership price p. 2. At what price should you set the membership fee to obtain the maximum revenue? What is the maximum possible revenue? 3. At what price should you set the membership fee to obtain the maximum profit? What is the maximum possible profit? What is the corresponding revenue?

13. Solution The annual revenue is given by The annual cost as function of q is given by The annual cost as function of p is given by

14. Solution Thus the annual profit function is given by

15. The graph of the revenue function is

16. The graph of the revenue function is

17. The profit function is

18. The profit function is

19. Vertex Form of a Parabola

20. Vertex Form of a Parabola To get the vertex form of the parabola we complete the square in x as indicated in the next steps:

21. Standard form Vertex form Vertex Form of a Parabola

22. Vertex Form of a Parabola Example: Find the vertex form of

23. Vertex Form of a Parabola Use the vertex form of to graph the parabola

24. Vertex Form of a Parabola Use the vertex form of to graph the parabola

25. Find a Quadratic Function Given Its Vertex and One Other Point

26. Vertex Form of a Parabola Determine the quadratic function whose vertex is (2, 3) and whose y-intercept is 1.

28. More Examples

29. Maximizing Revenue • The marketing department at Widgets Inc. found that, when certain • widgets are sold at a price of p dollars per unit, the number x of • widgets sold is given by the demand equation • x = 1500  30p • Find a model that expresses the revenue R as a function of the price p. • What is the domain of R? • What unit price should be used to maximize revenue? • If this price is charged, what is the maximum revenue?

30. Maximizing Revenue • The marketing department at Widgets Inc. found that, when certain • widgets are sold at a price of p dollars per unit, the number x of • widgets sold is given by the demand equation • x = 1500  30p • Find a model that expresses the revenue R as a function of the price p. • What is the domain of R?

31. Maximizing Revenue • The marketing department at Widgets Inc. found that, when certain • widgets are sold at a price of p dollars per unit, the number x of • widgets sold is given by the demand equation • x = 1500  30p • What unit price should be used to maximize revenue? • If this price is charged, what is the maximum revenue?

32. Maximizing Revenue • The marketing department at Widgets Inc. found that, when certain • widgets are sold at a price of p dollars per unit, the number x of • widgets sold is given by the demand equation • x = 1500  30p • How many units are sold at this price? • Graph R. • What price should Widgets Inc. charge to collect at least $12,000 in revenue?

33. Maximizing Revenue The marketing department at Widgets Inc. found that, when certain widgets are sold at a price of p dollars per unit, the number x of widgets sold is given by the demand equation x = 1500  30p So the company should charge between $10 and $40 to earn at least $12,000 in revenue.

34. Maximizing Area A farmer has 1600 yards of fence to enclose a rectangular field. What are the dimensions of the rectangle that encloses the most area? The farmer should make the rectangle 400 yards by 400 yards to enclose the most area.

35. Projectile Motion 1. Find the maximum height of the projectile.

36. Projectile Motion 2. How far from the base of the cliff will the projectile strike the water? Solution cannot be negative so the projectile will hit the water about 5458 feet from the base of the cliff.

37. The Golden Gate Bridge

38. The Golden Gate Bridge

39. The Golden Gate Bridge