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2-1 QUADRATIC FUNCTIONChapter 2
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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.
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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
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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 .
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EXAMPLES OF QUADRATIC FUNCTIONS Lets just graph the examples :
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PROPERTIES OF THE QUADRATIC FUNCTION
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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.
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IDENTIFYING THE VERTEX OF A QUADRATIC FUNCTION
Write the quadratic function given by in standard form and find the vertex of the graph.
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IDENTIFYING THE VERTEX OF A QUADRATIC FUNCTION
Write the quadratic function given by in standard form and find the vertex of the graph.
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STUDENT GUIDED PRACTICE
Do problems 23-25 from book page 96
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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
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EXAMPLE OF FINDING THE X-INTERCEPTS Find the x intercepts for the graph of
each function given below a) b) )
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STUDENT GUIDED PRACTICE
Do problems 49 and 51 from book page 97
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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.
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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.
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CONTINUE
Step 3 Then find the y-value of the vertex,
The maximum value is -11/3
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STUDENT GUIDED PRACTICE
Find the minimum or maximum value of f(x) = 6x2 + 5x – 4.
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STUDENT GUIDED PRACTICE
Find the maxima or minima from the following function
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HOMEWORK
Do problems 24-27 and 33 and 36 from page 96
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CLOSURE
Today we learned about quadratic equations
Next time we are going to continue with 2.3