Algebra II/Trig Honors
Unit 2 Day 2: Evaluate and Graph Polynomial Functions
Objective: To evaluate and graph polynomial functions.
Big Ideas: Algorithms are developed to make tedious work easier.
Graphs are sets of paired data, possibly generated from an equation.
Features of an equation give us clues about characteristics of their graphs.
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