1. BUS-221 Quantitative Methods
LECTURE 2

2. Learning Outcome
Knowledge - Be familiar with basic mathematical techniques including: systems of linear equations
Research - Retrieve and analyse information from directed sources for calculation and interpretation
 Mentation - Analyse business case studies and make decisions based on quantitative data.

3. Topics
Functions ▪ quadratic equations
Functions ▪ graphing functions

4. Quadratic Equations (1 of 9)
A quadratic equation is also called a second-degree equation or an equation of degree two.

5. Quadratic Equations (2 of 9)
Example 1 – Solving a Quadratic Equation by Factoring

6. Quadratic Equations (3 of 9)
Example 1 – Continued

7. Quadratic Equations (4 of 9)
Example 2 – Solving a Higher-Degree Equation by Factoring

8. Quadratic Equations (5 of 9)
Example 3 – Solution by Factoring

9. Quadratic Equations (6 of 9)
Quadratic Formula

10. Quadratic Equations (7 of 9)
Example 4 – A Quadratic Equation with One Real Root

11. Quadratic Equations (8 of 9)
Quadratic-Form Equation
When an equation that is not quadratic can be transformed into a quadratic equation by an appropriate substitution, the given equation is said to have quadratic-form.
Example 5 – Solving a Quadratic-Form Equation

12. Quadratic Equations (9 of 9)
Example 6 – Continued
If this PowerPoint presentation contains mathematical equations, you may need to check that your computer has the following installed:
1) Math Type Plugin
2) Math Player (free versions available)
3) NVDA Reader (free versions available)

13. Functions (1 of 5)
A function assigns each input number to one output number.
The set of all input numbers is the domain of the function.
The set of all output numbers is the range.

14. Functions (2 of 5) Example 1 – Determining Equality of Functions
Determine which of the following functions are equal.

15. Functions (3 of 5)
Example 1 – Continued

16. Functions (4 of 5)
Example 2 – Finding Domain and Function Values

17. Functions (5 of 5)
Example 3 – Demand Function

18. Special Functions (1 of 4)
Example 1 – Constant Function

19. Special Functions (2 of 4)
Example 2 – Rational Functions
Example 3 – Absolute-Value Function

20. Special Functions (3 of 4)

21. Special Functions (4 of 4)
Example 4 – Genetics

22. Combinations of Functions (1 of 5)

23. Combinations of Functions (2 of 5)
Example 1 – Combining Functions

24. Combinations of Functions (3 of 5)
Example 1 – Continued
We can also combine two functions by first applying one function to an input and then applying the other function to the output of the first.
This is called composition.

25. Combinations of Functions (4 of 5)

26. Combinations of Functions (5 of 5)
Example 2 – Composition

27. Inverse Functions (1 of 4)

28. Inverse Functions (2 of 4)
Example 1 – Inverses of Linear Functions

29. Inverse Functions (3 of 4)
Example 2 – Inverses Used to Solve Equations

30. Inverse Functions (4 of 4)
Example 3 – Finding the Inverse of a Function

31. Quadratic Functions (1 of 5)

32. Quadratic Functions (2 of 5)
Example 1 – Graphing a Quadratic Function

33. Quadratic Functions (3 of 5)
Example 2 – Graphing a Quadratic Function

34. Quadratic Functions (4 of 5)
Example 3 – Finding and Graphing an Inverse

35. Quadratic Functions (5 of 5)
Example 3 – Continued

36. Graphs in Rectangular Coordinates (1 of 5)
A rectangular coordinate system allows us to specify and locate points in a plane. It also provides a geometric way to graph equations in two variables.

37. Graphs in Rectangular Coordinates (2 of 5)
Example 1 – Intercepts of a Graph

38. Graphs in Rectangular Coordinates (3 of 5)
Example 2 – Intercepts of a Graph

39. Graphs in Rectangular Coordinates (4 of 5)
Example 3 – Graph of the Absolute-Value Function

40. Graphs in Rectangular Coordinates (5 of 5)
Example 4 – Graph of a Case-Defined Function
Graph the case-defined function
Solution:

41. Symmetry (1 of 5)
Example 1 – y-axis Symmetry

42. Symmetry (2 of 5)

43. Symmetry (3 of 5)
Example 2 – Graphing with Intercepts and Symmetry

44. Symmetry (4 of 5)
Example 3 – Continued

45. Symmetry (5 of 5)
Example 4 – Symmetry about the Line y = x

46. Translations and Reflections (1 of 3)
Some functions and their associated graphs occur so frequently that we find it worthwhile to memorize them. Below are six such basic functions.

47. Translations and Reflections (2 of 3)
The table below gives a list of basic types of transformations:
Table 2.2 Transformations, c > 0
Equation
How to Transform Graph of y =f(x) to Obtain Graph of Equation
y = f (x) + c
shift c units upward
y = f (x) − c
shift c units downward
y = f (x − c)
shift c units to right
y = f (x + c)
shift c units to left
y = −f (x)
reflect about x-axis
y = f (−x)
reflect about y-axis
y = cf (x) c > 1
vertically stretch away from x-axis by a factor of c
y = cf (x) c < 1
vertically shrink toward x-axis by a factor of c

48. Translations and Reflections (3 of 3)
Example 1 – Horizontal Translation

49. Functions of Several Variables (1 of 7)

50. Functions of Several Variables (2 of 7)
Example 1 – Functions of Two Variables

51. Functions of Several Variables (3 of 7)
Example 2 – Temperature-Humidity Index

52. Functions of Several Variables (4 of 7)

53. Functions of Several Variables (5 of 7)

54. Functions of Several Variables (6 of 7)
Example 3 – Sketching a Surface

55. Functions of Several Variables (7 of 7)
Example 4 – Sketching a Surface