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-intercept There 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-intercept The x-intercept of a quadratic equation is determined by finding the roots of an equation. Finding roots by factorisation If 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 formula The 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

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