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