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6.4 Polynomial Functions
• Polynomial in one variable : A polynomial with only one variable
• Leading coefficient: the coefficient of the term with the highest degree in a polynomial in one variable
• Polynomial Function: A polynomial equation where the y is replaced by f(x)
State the degree and leading cofficient of each polynomial, if it is not a polynomial in
one variable explain why.1. 7x4 + 5x2 + x – 9
2. 8x2 + 3xy – 2y2
3. 7x6 – 4x3 + x-1
4. ½ x2 + 2x3 – x5
Evaluating Functions
• Evaluate f(x) = 3x2 – 3x +1 when x = 3
• Find f(b2) if f(x) = 2x2 + 3x – 1
• Find 2g(c+2) + 3g(2c) if g(x) = x2 - 4
End Behavior
• Describes the behavior of the graph f(x) as x approaches positive infinity or negative infinity.
• Symbol for infinity
End behavior Practice
f(x) - as x +
f(x) - as x -
End Behavior Practice
f(x) + as x +
f(x) - as x -
End Behavior Practice
f(x) + as x +
f(x) + as x -
The Rules in General
To determine if a function is even or odd
• Even functions: arrows go the same direction• Odd functions: arrows go opposite directions
To determine if the leading coefficient is positive or negative
• If the graph goes down to the right the leading coefficient is negative
• If the graph goes up to the right then the leading coeffiecient is positive
The number of zeros • zeros are the same as roots: where the graph crosses the
x-axis– The number of zeros of a function can be equal to the exponent
or can be less than that by a multiple of 2.• Example a quintic function, exponent 5, can have 5, 3 or 1 zeros
• To find the zeros you factor the polynomial
Critical Points• points where the graph changes direction.
– These points give us maximum and minimum values• Relative Max/Min
Put it all together• For the graph given
– Describe the end behavior
– Determine whether it is an even or an odd degree
– Determine if the leading coefficient is positive or negative
– State the number of zeros
Cont…• For the graph given
– Describe the end behavior– Determine whether it is an
even or an odd degree– Determine if the leading
coefficient is positive or negative
– State the number of zeros
• For the graph given – Describe the end
behavior– Determine whether it
is an even or an odd degree
– Determine if the leading coefficient is positive or negative
– State the number of zeros
• For the graph given – Describe the end
behavior– Determine whether it is
an even or an odd degree
– Determine if the leading coefficient is positive or negative
– State the number of zeros