MTH108 Business Math I

Lecture 12

Chapter 6

Quadratic and Polynomial Functions

Objectives

• Generally, introduce the reader to nonlinear functions

• More specifically, provide an understanding of the algebraic and graphical characteristics of quadratic and polynomial functions

• Illustrate a variety of applications of these types of functions

Review

• Linear functions of one variable, two variables, more than two variables

• Applications; revenue function, cost function, profit function, demand function, supply function, market equilibrium

• Break-even point with graphical representation

Today’s Topics

• Quadratic functions• Characteristics of quadratic functions• Graphical representation

• Need So far, we have focused on linear and non linear

mathematics and linear mathematics is very useful and convenient.

There are many phenomena which do not behave in a linear manner and can not be approximated by using linear functions.

For this purpose, we need to introduce nonlinear functions.

One of the more common nonlinear function is the quadratic function.

Quadratic Functions

Definition A quadratic function involving one independent variable x and the dependent variable y has the general form

Observe that the coefficient of can not be equal zero.Clearly, if then our equation reduces to

Examples

As long as can take any value. We will see this from the following examples.

Examples 1)

Examples (contd.)

2)

3)

Examples (contd.)

4)

5)

Examples (contd.)

6)

7)

Note that in chapter 4 we have given the general form of the quadratic function as

Here we have given the general form as

Both the general forms are definitely equivalent as we have only renamed the constants.

Graphical Representation

Recall that all linear functions are graphed as straight lines. What would be the graph of quadratic functions?

A straight line or a curve?All quadratic functions have graphs as curves called the

parabolas.e.g. consider the function

Properties of Parabolas• A parabola which opens “upward” is said to be

concave up.• A parabola which opens “downward” is said to be

concave down.• The point at which a parabola either bottoms out or

peaks out is called the vertex of the parabola.

Properties of Parabolas• Given a quadratic function of the general form, the

coordinates of the vertex of the parabola are

Properties of Parabolas• A parabola is a curve having a particular symmetry.• The line which passes through the vertex is called the

axis of symmetry.

• This line separates the parabola into two equal halves.

Sketching of Parabola

Parabolas can be sketched by using the method of chapter 4. But, there are certain things which can make the sketching relative easy.

These include:• Concavity of the parabola• Y-intercept• X-intercept• Vertex

Concavity

Concavity of a parabola, when a function is given in the general form can be determined by the sign of the coefficient on the term.

Examples

1)

2)

3)

4)

Intercepts

Recall that the y-intercept of a function is a point at which the graph intersects the y-axis.

In particular, when

Given the general form of the quadratic function

Examples

1)

2)

3)

4)

Intercepts

Recall that the x-intercept of a function is a point at which the graph intersects the x-axis.

In particular, when

Given the general form of the quadratic function

Methods to find the x-interceptThere are number of ways to find the x-intercepts.For quadratic functions, there may be one x-intercept,

two intercepts or no intercept.

Methods to find the x-interceptThe x-intercept of a quadratic equation is determined

by finding the roots of an equation.Finding roots by factorisationIf a quadratic can be factored, it is an easy way to find

the roots. E.g.

Recall that there are only three possibilities of the roots of a second degree equation.

Finding roots by using the quadratic formulaThe quadratic formula will always identify the real roots

of an equation if any exist.The quadratic formula of an equation which has the

general form

will be

Cases

Vertex

The vertex of a parabola can be found by using the formula.

When an equation has two intercepts, the vertex lies midway between the two x-intercepts.

When an equation has one intercept, the vertex lies on the intercept.

Example

Example

Alternatively,

Example

Vertex using x-intercepts:

Sketching a quadratic function

Summary

• Quadratic functions• Graph of quadratic functions• Concavity• Intercepts• Vertex

• Section 6.1 Q.1-35