Algebra 2
5.2 Notes
5.2 (Day One) Graphing Polynomial Functions
Date: ___________
What is a Polynomial Function?
Learning Target C: I can write a polynomial function in standard form and identify its degree and
leading coefficient.
We have talked about linear, quadratic, and cubic functions, and all of these are examples of
polynomial functions, which are categorized by their degree. A linear function has a degree of 1, a
quadratic is of degree 2, and a cubic is a degree 3 polynomial function.
Standard form of a Polynomial Function of Degree n
𝑝(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0
Where 𝑎𝑛 , 𝑎𝑛−1 , … , 𝑎2 , 𝑎1 , 𝑎𝑛𝑑 𝑎0 are real number coefficients
**Terms must be in order by their exponents, starting with the highest and ending with
the lowest**
Degree of a Polynomial: the ___________________ exponent when the polynomial is written in
standard form.
Leading Coefficient: the coefficient of the _____________ term when the polynomial is written in
standard form.
Write each polynomial function in standard form. Then, identify its degree and leading
coefficient.
A. 𝑓(𝑥) = 𝑥 3 + 4𝑥 2 − 𝑥 4 + 1
B. 𝑝(𝑥) = 𝑥 + 9𝑥 3 − 2𝑥 + 6𝑥 2
Intercept Form of a Polynomial Function
𝑝(𝑥) = 𝑎(𝑥 − 𝑥1 )(𝑥 − 𝑥2 ) … (𝑥 − 𝑥𝑛 )
Where 𝑎, 𝑥1 , 𝑥2 , … , 𝑎𝑛𝑑 𝑥𝑛 are real numbers
The polynomial has degree n, where n is the number of variable factors.
Given each function in Intercept form, write it in standard form, and identify the degree and
leading coefficient.
A. 𝑓(𝑥) = 𝑥 2 (𝑥 + 1)
B. 𝑓(𝑥) = −3𝑥(𝑥 − 1)(𝑥 + 2)2
1
Algebra 2
5.2 Notes
Investigating the End Behavior of Simple Polynomial Functions
Learning Target D: I can determine the end behavior of a polynomial function from its degree and
leading coefficient.
Relating End Behavior with Degree
Graph the following functions on a graphing calculator to discover the relationship between the degree of a polynomial
with its end behavior.:
𝑓(𝑥) = 𝑥, 𝑓(𝑥) = 𝑥 2 , 𝑓(𝑥) = 𝑥 3 , 𝑓(𝑥) = 𝑥 4 , 𝑓(𝑥) = 𝑥 5 , 𝑎𝑛𝑑 𝑓(𝑥) = 𝑥 6
Relating End Behavior with Leading Coefficient
Compare the graphs of the following functions with the graphs from the functions above to determine the relationship
between the leading coefficient of a polynomial and its end behavior:
𝑓(𝑥) = −𝑥, 𝑓(𝑥) = −𝑥 2 , 𝑓(𝑥) = −𝑥 3 , 𝑓(𝑥) = −𝑥 4 , 𝑓(𝑥) = −𝑥 5 , 𝑎𝑛𝑑 𝑓(𝑥) = −𝑥 6
Fill in the table with your findings:
Type of Function
End Behavior with Positive
Leading Coefficient
End Behavior with Negative
Leading Coefficient
Even Degree
Odd Degree
Given each graph, tell whether the degree of the function is even or odd and identify whether
the leading coefficient is positive or negative.
A.
B.
C.
Degree:
Degree:
Degree:
LC:
LC:
LC:
2
Algebra 2
5.2 Notes
Investigating the Turning Points of the Graphs of Polynomial Functions
Recall: A turning point is a point where the graph changes from increasing to decreasing or
decreasing to increasing. Turning points result in local minimum or local maximum values.
Maximum and Minimum Values
Global
The function never takes on a value that is
greater than the maximum or less than the
minimum
Local
A maximum or minimum within some interval
around the turning point that does not need to
be (but may be) a global maximum or global
minimum
Example) Given the graph, determine the number of turning points, the
number of global maximum and/or minimum values, and the number of
local maximum and/or minimum values that are not global.
For each graph, tell the number of turning points and the number of Global max/min and/or
local max/min values that are not global.
A.
B.
C.
Turning Points: _______
Turning Points: _______
Turning Points: _______
Global Max: ___________
Global Max: ___________
Global Max: ___________
Local Max: ____________
Local Max: ____________
Local Max: ____________
Global Min: ___________
Global Min: ___________
Global Min: ___________
Local Min: ____________
Local Min: ____________
Local Min: ____________
3
Algebra 2
5.2 Notes
Finding the Zeros of a Polynomial Function
Learning Target E: I can find the zeros of a polynomial function in intercept form.
Recall that the zeros of a function give the graph’s ___________________________. Finding the zeros of a
polynomial function is easiest when the function is in intercept form. All we need to do is:
______________________________________________________________________.
Find the x-intercepts and state the degree of each polynomial function. (x-intercepts are
always written as ordered pairs, (x, 0) )
A. f ( x)  x( x  3)( x  1)
B. f ( x)  ( x  4) 2 ( x  1)( x  1)
C. f ( x)  ( x  2) 2 ( x  6)
D. 𝑝(𝑥) = 𝑥(𝑥 + 5)3
5.2 (Day Two) Graphing Polynomial Functions
Date: _____________
Investigating the Behavior of the Graph of a Polynomial Function at Its Zero Values
Notice some of the factors in the functions above had exponents other than 1, meaning they occur
more than once. The number of times a factor occurs is called its multiplicity.
Let’s see how the multiplicity of a factor affects the behavior of the graph at its related xintercept! Graph each function on a graphing calculator, and sketch them below. Be sure to
accurately plot the x-intercepts.
A. f ( x)  x( x  3)( x  1)
B. f ( x)  ( x  4) 2 ( x  1)( x  1)
4
Algebra 2
5.2 Notes
C. f ( x)  ( x  2) 2 ( x  6)
D. 𝑝(𝑥) = 𝑥(𝑥 + 5)3
Use what you discovered to fill in the table below:
Behavior at x-intercepts
Goes Straight Through
Tangent to x-axis (“Bounces”)
“Squiggles” Through
Sketching the Graph of a Polynomial Function in Intercept Form
Learning Target F: I can use end behavior, x-intercepts, and the y-intercept to graph a polynomial
function in intercept form.
Sketch the graph of each polynomial function. Identify the x- and y- intercepts.
A. 𝑓(𝑥) = 𝑥(𝑥 + 2)(𝑥 − 3)
B. 𝑓(𝑥) = −(𝑥 − 4)(𝑥 − 1)(𝑥 + 1)(𝑥 + 2)
5
Algebra 2
5.2 Notes
C. 𝑓(𝑥) = −𝑥 2 (𝑥 − 4)
D. 𝑓(𝑥) = (𝑥 − 2)(𝑥 + 1)(𝑥 + 4)2
Writing Polynomial Functions from their Graphs
Learning Target G: I can write a polynomial function from its graph.
Write the equation of each graph in intercept form, with integer x-intercepts. Assume the
leading coefficient, a, is either 1 or -1.
A.
B.
Identify zeros and multiplicities:
Identify zeros and multiplicities:
Degree:
Degree:
Leading Coefficient:
Leading Coefficient:
Equation:
Equation:
6
Algebra 2
C.
5.2 Notes
D.
Identify zeros and multiplicities:
Identify zeros and multiplicities:
Degree:
Degree:
Leading Coefficient:
Leading Coefficient:
Equation:
Equation:
7