Download 10-04 NOTES Characteristics of Polynomial Functions and Evaluate

2.2 Polynomial Functions.notebook
October 04, 2016
2.2 Polynomial Functions
Definition:
A polynomial function in one variable has the form
f(x) = anxn + an­1 xn­1 + .... + a 1x1 + a0,
where all the exponents are positive integers and the coefficients are real numbers.
EXAMPLES:
8x5 ­ 6x4 ­ 2x2 + x ­ 9
3x4 + 5x When written in descending order:
an is the leading coefficient,
a0 is the constant,
n is the degree of the polynomial Oct 20­2:38 PM
Classify Polynomials by Degree:
Degree
Name
Example
0
1
2
3
4
5
PREVIOUS EXAMPLES: Classify each polynomial by degree. 1. 3x + 9x 2 + 5 2. 2x3 ­ x + 5x4 3. 3 ­ 4x5 + 9x2 + 10 Oct 20­2:38 PM
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2.2 Polynomial Functions.notebook
October 04, 2016
End Behavior of Polynomial Functions
Even Degree Polynomials
Look at these graphs. What do you observe about the ends
if the degree is even?
positive leading
coefficient
negative leading
coefficient
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Oct 20­2:38 PM
End Behavior of Polynomial Functions
Odd Degree Polynomials
Look at these graphs. What do you observe about the ends
if the degree is odd?
positive leading
coefficient
negative leading
coefficient
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Oct 20­2:38 PM
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2.2 Polynomial Functions.notebook
October 04, 2016
Summary: End Behavior of Polynomial Functions
Most important thing to consider:
the sign and degree of the leading term.
When the leading coefficient is positive, the graph __________________ to the right.
When the leading coefficient is negative, the graph __________________ to the right.
When the function's degree is odd, the ends go in __________________ directions.
When the function's degree is even, the ends go in __________________ directions.
Oct 20­2:38 PM
Practice: End Behavior of Polynomial Functions
REMEMBER!
Most important thing to consider?
Example 1.
Which of the following could be the graph of a polynomial whose leading term is ­3x4 ?
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Oct 20­2:38 PM
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2.2 Polynomial Functions.notebook
October 04, 2016
Practice: End Behavior of Polynomial Functions
REMEMBER!
Most important thing to consider?
Example 2.
Describe the end behavior of 4x4 + 2x3 ­ x + 7.
Example 3.
Describe the end behavior of 3x2 + 6x ­ x3 ­ 2.
Example 4.
Describe the end behavior of 3x7 + 5x + 1004.
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Practice End Behavior
Turning Points of Polynomial Functions
points where the graph changes direction, the BUMPS.
A linear function has no turning point.
A quadratic function has only 1 turning point.
A cubic function has at most 2 turning points.
A quartic function has at most 3 turning points.
Summary: A polynomial function has at most
_________________ turning points.
Turning Points
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2.2 Polynomial Functions.notebook
October 04, 2016
PRACTICE: For each graph, determine the sign of the
leading coefficient and the least possible degree of
the polynomial function.
1.
2.
3.
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Mar 7­8:06 AM
EXAMPLES:
For each polynomial function, state the degree, describe the end behavior, and sketch the general shape.
1. f(x) = 5 ­ x 2. f(x) = x3 ­ 4x
3. f(x) = x4 ­ 2x2 + 2 4. f(x) = 4x ­ x2 ­ 1
5. f(x) = 3x3 ­ 3x ­ x4 ­ 3 6. f(x) = 6x + 1 ­ x3
Oct 20­7:22 PM
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2.2 Polynomial Functions.notebook
October 04, 2016
II. Evaluate Polynomial Function
A. Direct Substitution: Substitute the given value of
x into the function and find the value of y.
EXAMPLES:
2
1. f(x) = 2x + 6x - 8, find f(2).
3
2
2. f(x) = -x + 4x + 37, find f(-3).
2
3. f(x) = x - 5x, find f(-4).
2
4. f(x) = 2x - 5x + 1, find f(¾).
Oct 20­2:38 PM
2.2 Polynomial Functions
B. Synthetic Substitution
STEPS:
EXAMPLE:
3
2
1. f(x) = 5x + 3x - x + 7, find f(2).
1. Write the coefficients in order of
descending exponents. Include a
coefficient of 0 for any missing term.
2. Write the value for x to the left.
3. Bring down the leading coefficient.
4. Multiply the leading coefficient with
the x value to the left. Write the
product under the second coefficient.
Add.
5. Multiply this sum with the x value.
Write under the third coefficient.
Add.
6. Repeat for the remaining coefficents.
7. The final sum is the value of the
function at the given value of x.
Oct 20­2:38 PM
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2.2 Polynomial Functions.notebook
October 04, 2016
B. Synthetic Substitution
EXAMPLE:
4
3
STEPS:
2. f(x) = -2x - x + 4x - 5 for x = - 1.
1. Write the coefficients in order of
descending exponents. Include a
coefficient of 0 for any missing term.
2. Write the value for x to the left.
3. Bring down the leading coefficient.
4. Multiply the leading coefficient with
the x value to the left. Write the
product under the second coefficient.
Add.
5. Multiply this sum with the x value.
Write under the third coefficient.
Add.
6. Repeat for the remaining coefficents.
7. The final sum is the value of the
function at the given value of x.
Oct 20­2:38 PM
Homework
page 99, problems 3-8 all, 9-21 odd,
24-33 all
Oct 20­2:38 PM
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Attachments
Explore Graphs of Common Polynomial Functions.doc