1. 1 Linear and Quadratic Functions On completion of this module you should be able to: define the terms function, domain, range, gradient, independent/dependent variable use function notation recognise the relationship between functions and equations graph linear and quadratic functions calculate the function given initial values (gradient, 1 or 2 coordinates) solve problems using functions model elementary supply and demand curves using functions and solve associated problems

2. 2 A function describes the relationship that exists between two sets of numbers. Put another way, a function is a rule applied to one set of numbers to produce a second set of numbers. Functions

3. 3 Example: Converting Fahrenheit to Celsius This rule operates on values of F to produce values of C. The values of F are called input values and the set of possible input values is called the domain. The values of C are called output values and the set of output values produced by the domain is called the range.

4. 4 Consider the function The x are the input values and f(x), read f of x, are the output values. The domain is the set of positive real numbers including 0 and excepting 3. (Why?) The output values produced by the domain is the range. Sometimes the symbol y is used instead of f(x). Function Notation

5. 5 An equation is produced when a function takes on a specific output value. eg f(x) = 3x + 6 is a function. When f(x) = 0, then the equation becomes 0 = 3x + 6 which can be easily solved (to give x = −2) Function and Equations

6. 6 This is shown graphically as follows:

7. 7 Input and output values form coordinate pairs: (x, f(x)) or (x, y). x values measure the distance from the origin in the horizontal direction and f(x) values the distance from the origin in the vertical direction. To plot a straight line (linear function), 2 sets of coordinates (3 sets is better) must be calculated. For other functions, a selection of x values should be made and coordinates calculated. Graphing Functions

8. 8 Graph f(x) = 2x − 4 Example: Linear Function

9. 9

10. 10 At the y-intercept, x = 0, so and the coordinate is (0,2). Example: Quadratic Function Graph the function:

11. 11 At the x-intercept, f(x) = 0, so and the coordinates are (2,0) and (0.5,0).

12. 12 Vertex: The coordinates of the vertex are: (1.25, −1.125).

13. 13 1 2 2 x (1.25, -1.125) (2,0) (0.5,0) (0,2)

14. 14 All linear functions (or equations) have the following features: a slope or gradient (m) a y-intercept (b) If (x 1, y 1 ) and (x 2, y 2 ) are two points on the line then the gradient is given by: Linear Functions

15. 15 Gradient is a measure of the steepness of the line. If m > 0, then the line rises from left to right. If m < 0, the line falls from left to right. A horizontal line has a gradient of 0; a vertical line has an undefined gradient. The y-intercept is calculated by substituting x = 0 into the equation for the line.

16. 16 All straight line functions can be expressed in the form y = mx + b Note: The standard form equation for linear functions is Ax + By + C = 0. Equations in this form are not as useful as when expressed as y = mx + b. Equations can be derived in the following way, depending on what information is given.

17. 17 1.Given (x 1, y 1 ) and (x 2, y 2 ) 2.Given m and (x 1, y 1 ) 3.Given m and b Deriving Straight Line Functions

18. 18 A tractor costs $60,000 to purchase and has a useful life of 10 years. It then has a scrap value of $15,000. Find the equation for the book value of the tractor and its value after 6 years. Problem: Depreciation

19. 19 V ? 6 15,000 10 t 60,000

20. 20 Value (V) depends on time (t). t is called the independent variable and V the dependent variable. The independent variable is always plotted on the horizontal axis and the dependent variable on the vertical axis.

21. 21 Given two points, the equation becomes:

22. 22 The book value of the tractor after 6 years is $33,000. or more correctly

23. 23 Suppose a manufacturer of shoes will place on the market 50 (thousand pairs) when the price is $35 (per pair) and 35 (thousand pairs) when the price is $30 (per pair). Find the supply equation, assuming that price p and quantity q are linearly related. Example

24. 24

25. 25 The supply equation is

26. 26 For sheep maintained at high environmental temperatures, respiratory rate r (per minute) increases as wool length l (in centimetres) decreases. Suppose sheep with a wool length of 2cm have an (average) respiratory rate of 160, and those with a wool length of 4cm have a respiratory rate of 125. Assume that r and l are linearly related. (a) Find an equation that gives r in terms of l. (b) Find the respiratory rate of sheep with a wool length of 1cm. Example

27. 27 (a) Find r in terms of l  l is independent r is dependent Coordinates will be of the form: (l, r).

28. 28

29. 29 When wool length is 1 cm, average respiratory rate will be 177.5 per minute. (b) When l = 1

30. 30 All quadratic functions can be written in the form where a, b and c are constants and a  0. Quadratic Functions

31. 31 In general, the higher the price, the smaller the demand is for some item and as the price falls demand will increase. Demand curve q p Elementary Supply and Demand

32. 32 Concerning supply, the higher the price, the larger the quantity of some item producers are willing to supply and as the price falls, supply decreases. q p Supply curve

33. 33 Note that these descriptions of supply and demand imply that they are dependent on price (that is, price is the independent variable) but it is a business standard to plot supply and demand on the horizontal axis and price on the vertical axis.

34. 34 The supply of radios is given as a function of price by and demand by Find the equilibrium price. Example: Equilibrium price

35. 35 Graphically, Note the restricted domains. 12345 p 70 equilibrium price 0 0

36. 36 Algebraically, D(p) = S(p)

37. 37 −14 is not in the domain of the functions and so is rejected. The equilibrium price is $4.

38. 38 If an apple grower harvests the crop now, she will pick on average 50 kg per tree and will receive $0.89 per kg. For each week she waits, the yield per tree increases by 5 kg while the price decreases by $0.03 per kg. How many weeks should she wait to maximise sales revenue? Example: Maximising profit

39. 39 Weight and Price can both be expressed as functions of time (t). W(t) = 50 + 5t P(t) = 0.89 − 0.03t

40. 40 Maximum occurs at She should wait 9.83 weeks ( 10 weeks) for maximum revenue. (R = $59 per tree)