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Algebra II/Trig Honors
Unit 2 Day 2: Evaluate and Graph Polynomial Functions
Objective: Graph polynomial functions
Definitions:

Polynomial - _______________________________________________________________

Polynomial Function - ________________________________________________________
where a0  0 , exponents are positive whole numbers, and coefficients are all real numbers.

o
a n and is called the _____________________________________
o
n is the _____________________________
o
a 0 is the ____________________________
Standard Form of Polynomial Functions - ________________________________________
__________________________________________________________________________
Example 1: Identifying Polynomial Functions
Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type,
and leading coefficient.
a. h x   x 4 
1 2
x 3
4
c. f x   5x 2  3x 1  x
b. g x   7 x  3  x 2
d. k x   x  2 x  0.6 x 5
Example 2: Evaluate by Direct Substitution
Use direct substitution to evaluate f x   2 x 4  5x 3  4 x  8 when x  3

Another way to evaluate a polynomial function is to use ___________________________.
o This method requires fewer operations than direct substitution.
Example 3: Synthetic Substitution
Use synthetic substitution to evaluate f x   2 x 4  5x 3  4 x  8 when x  3 . Your answer should match the
answer in the previous example.
1. Write the coefficients of f x  , include zero as
a placeholder if necessary, in order of descending
exponents. Write the value at which f x  is
being evaluated to the left.
2. Bring down the leading coefficient. Multiply
the leading coefficient by the x-value. Write the
product under the second coefficient. Add.
3. Multiply the previous sum by the x-values.
Write the product under the third coefficient.
Add. Repeat for all of the remaining coefficients.
The final sum is the value of f x  at the given
value.

End Behavior - _______________________________________________________________
____________________________________________________________________________
o For polynomial functions, the end behavior is determined by the function’s degree and the sign
of its leading coefficients.
**The expression x   is read as “x approaches positive infinity”
Example 4: End Behavior
What is true about the degree and leading coefficient of the polynomial function whose graph is shown?
Degree is odd or even? ___________________
Leading coefficient is positive or negative? ___________________
Actual number of turns? ________________________
Number of solutions? ______ Number of zeros? _________
Number of real solutions? _______
Number of complex solutions? __________
Number of x – intercepts? _______

The maximum possible number of turns can be found by subtracting one from the degree of the
polynomial.
Steps for Graphing Polynomial Functions:
1. Determine end behavior of the function.
2. Determine number of solutions of the function.
3. Determine maximum possible number of turns.
4. Make a table of values and plot the corresponding points. (Use synthetic substitution if the problem is a
no calculator problem.)
5. Connect the points with a smooth curve and check the end behavior.
6. Sometimes you can check solutions/zeros/x-intercepts by factoring the function.
7. If it’s a homework problem, you can confirm accuracy of graph by graphing with your calculator.
Example 5: Graph Polynomial Functions
a. Graph f x    x 3  x 2  3x  3
.
b. Graph f x   x 4  x 3  4 x 2  4
HW: Page 99 #3-8, 9-21 (M3), 25-35 odd, 39-48 (M3), 54
Algebra II/Trig Honors
Unit 2 Day 2: Evaluate and Graph Polynomial Functions
Class Practice (Day 1 and Day 2 Skills):
Evaluate the expression.
1.
4 
2 3
2.
2
3.  
9
 8 8
3
3
4.
6  10 4
9  10 7
Simplify the expression.
6
5
5. x x x
3

2
6. 7 y z
5
y
4
z
1

 s3
7.   4
t



2
 x 4 y 2 
8.  3 6 
 x y 
3
9. Rewrite the given polynomial in standard form. Identify the leading coefficient, degree, and number of
terms. Name the polynomial.
5x 3  2x  1  10x 2  9x 5  3x 4
Graph each polynomial function on a calculator. Describe the graph. Determine the end
behavior of the graph. Find the real zeros for each equation. Write the x – intercepts.
Write the y – intercept. Find the maximum and minimum point. Find the maximum and
minimum values at the maximum and minimum point.
f ( x )  2x 3  6 x  1
10.
11. f ( x )  5x 4  4x 3  5x  3
12. The profit, P, earned by a small business each year can be modeled to fit the polynomial function
P( y )  10y 3  50y 2  20y  100,000, where y is the number of years since 1990. Did the company’s profits
increase or decrease in 1995, compared to 1994?
Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type,
and leading coefficient.
11. f x  13  2x
12. px   9 x 4  5x 2  4
13. hx   6 x 2    3x
Use direct substitution to evaluate the polynomial function for the given value of x.
14. f x   x 4  2 x 3  3x 3  7 ; x  2
15. g x   x 3  5 x 2  6 x  1 ; x  4
Use synthetic substitution to evaluate the polynomial function for the given value of x.
16. f x   5 x 3  3x 2  x  7 ; x  2
17. g x   2 x 4  x 3  4 x  5 ; x  1
18. Describe the degree and leading coefficient of the polynomial function whose graph is shown.