BUS-221 Quantitative Methods LECTURE 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.

Topics Functions ▪ quadratic equations Functions ▪ graphing functions

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

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

Quadratic Equations (3 of 9) Example 1 – Continued

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

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

Quadratic Equations (6 of 9) Quadratic Formula

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

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

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)

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.

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

Functions (3 of 5) Example 1 – Continued

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

Functions (5 of 5) Example 3 – Demand Function

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

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

Special Functions (3 of 4)

Special Functions (4 of 4) Example 4 – Genetics

Combinations of Functions (1 of 5)

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

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.

Combinations of Functions (4 of 5)

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

Inverse Functions (1 of 4)

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

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

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

Quadratic Functions (1 of 5)

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

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

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

Quadratic Functions (5 of 5) Example 3 – Continued

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.

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

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

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

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

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

Symmetry (2 of 5)

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

Symmetry (4 of 5) Example 3 – Continued

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

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.

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

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

Functions of Several Variables (1 of 7)

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

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

Functions of Several Variables (4 of 7)

Functions of Several Variables (5 of 7)

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

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