chapter 2. a polynomial function has the form where are real numbers and n is a nonnegative...

2-1 QUADRATIC FUNCTIONChapter 2

WHAT IS A POLYNOMIAL FUNCTION?

A polynomial function has the form

where are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one or more monomials with real coefficients and nonnegative integer exponents. The degree of the polynomial function is the highest value for n where an is not equal to 0. Polynomial functions of only one term are called monomials or power functions.

CLASSIFICATION OF POLYNOMIALS

Polynomials are classify based on the leading exponent and that leading exponent is what we called degree.

has degree 0 and is called constant function.

has degree 1 and is called linear function

has degree 2 and is called quadratic function

QUADRATIC FUNCTION

quadratic function is a function of the form

where ,a, b and c and are real numbers and not equal to zero. The graph of the quadratic function is called a parabola. It is a "U" shaped curve that may open up or down depending on the sign of coefficient a .

EXAMPLES OF QUADRATIC FUNCTIONS Lets just graph the examples :

PROPERTIES OF THE QUADRATIC FUNCTION

THE STANDARD FORM OF A QUADRATIC FUNCTION

Any quadratic function can be written in the standard form

where h and k and are given in terms of coefficients a, b and c .

The graph of f is a parabola whose axis is the vertical line x=h and whose vertex is the point(h ,k ). When a>o, the parabola opens upward, and when a<0, the parabola opens downward.

IDENTIFYING THE VERTEX OF A QUADRATIC FUNCTION

Write the quadratic function given by in standard form and find the vertex of the graph.

IDENTIFYING THE VERTEX OF A QUADRATIC FUNCTION

Write the quadratic function given by in standard form and find the vertex of the graph.

STUDENT GUIDED PRACTICE

Do problems 23-25 from book page 96

IDENTIFYING THE X-INTERCEPTS OF THE QUADRATIC FUNCTION

The intercepts of the graph of a quadratic function given by

are the real solutions, if they exist, of the quadratic equation

EXAMPLE OF FINDING THE X-INTERCEPTS Find the x intercepts for the graph of

each function given below a) b) )

STUDENT GUIDED PRACTICE

Do problems 49 and 51 from book page 97

MAXIMUM AND MINIMUM

What is the minimum value? When the parabola opens upward , the

y-value of the vertex is the minimum value.

What is the maximum value? When the parabola opens downward

the y-value of the vertex is the maximum value.

FINDING THE MINIMUM AND MAXIMUM Find the minimum or maximum value of

f(x) = –3x2 + 2x – 4.

Solution:

Step 1 Determine whether the function has minimum or maximum value.

Because a is negative, the graph opens downward and has a maximum value.

Step 2 Find the x-value of the vertex.

CONTINUE

Step 3 Then find the y-value of the vertex,

The maximum value is -11/3

STUDENT GUIDED PRACTICE

Find the minimum or maximum value of f(x) = 6x2 + 5x – 4.

STUDENT GUIDED PRACTICE

Find the maxima or minima from the following function

HOMEWORK

Do problems 24-27 and 33 and 36 from page 96

CLOSURE

Today we learned about quadratic equations

Next time we are going to continue with 2.3