Algebra II/Trig Honors
Unit 2 Day 2: Evaluate and Graph Polynomial Functions
Objective:
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  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? ___________________

Graphing Polynomial Functions
o First plot points to determine the shape of the graph’s middle portion.
o Then use what you know about end behavior to sketch the ends of the graph.
Example 5: Graph Polynomial Functions
a. Graph f x    x 3  x 2  3x  3
1. Make a table of values and plot the corresponding points.
2. Connect the points with a smooth curve and check the
end behavior.
b. Graph f x   x 4  x 3  4 x 2  4
Example 6: The energy E (in foot-pounds) in each square foot of a wave is given by the model
E  0.0029 s 4 where s is the wind speed (in knots). Graph the model. Use the graph to estimate the
wind speed needed to generate a wave with 1000 foot-pounds of energy per square foot.
HW: Page 99 #3-8, 9-21 (M3), 25-35 odd, 39-48 (M3), 54