1. Zero’s, Multiplicity, and End Behaviors
Graphing Polynomials
Zero’s, Multiplicity, and End Behaviors

2. Defn:
A Polynomial Function
In the form of: 𝒇 𝒙 = 𝒂 𝒏 𝒙 𝒏 + 𝒂 𝒏−𝟏 𝒙 𝒏−𝟏 + ⋯ 𝒂 𝟏 𝒙+ 𝒂 𝟎
has coefficients that are real numbers,
exponents that are non-negative integers.
and a domain of all real numbers.
Are the following functions polynomials? no
𝑓 𝑥 =5𝑥+2 𝑥 2 −6 𝑥 3 +3
yes
𝑔 𝑥 =2 𝑥 2 −4𝑥+ 𝑥 −2
𝑘 𝑥 = 2 𝑥 𝑥 5 +3𝑥
ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥)
yes no

3. Determine which of the following are polynomial functions
Determine which of the following are polynomial functions. If the function is a polynomial, state its degree.
A polynomial of degree 4.
Not a polynomial because of the square root since the power is NOT an integer
Not a polynomial because of the x in the denominator since the power is negative

4. Graphs of polynomials are smooth and continuous.
No gaps or holes, can be drawn without lifting pencil from paper
No sharp corners or cusps
This IS NOT the graph of a polynomial
This IS the graph of a polynomial

5. Graphs of Polynomial Functions and Nonpolynomial Functions

6. Comprehensive Graphs
A comprehensive graph of a polynomial function will exhibit the following features: 1. all x-intercepts ( real zero’s if any) 2. the y-intercept 3. enough of the graph to reveal the correct end behaviors 4. all turning points, extreme points, and points of inflection 5. the behavior of the graph at the zero’s
(WHERE Y = 0)
(WHERE X = 0)
(DEGREE AND +/- a)
(n-1)
(MULTIPLICITY)

7. X-Intercepts

8. Defn: Real Zero of a function
If f(r) = 0 and r is a real number, then r is a real zero of the function.
Equivalent Statements for a Real Zero
r is a real zero of the function.
( r,0 ) is an x-intercept of the graph of the function.
x – r is a factor of the function.
r is a solution to the function f(x) = 0

9. r = -3, -2, -1, 0, 1, 2, 3 factors ( x + 3) ( x -1) ( x + 2) ( x - 2)
Use the x-intercepts to write the
factors of the function.
r = -3, -2, -1, 0, 1, 2, 3
factors
( x + 3)
( x -1) -3 -2 -1 1 2 3
( x + 2)
( x - 2)
( x + 1)
( x - 3) x

10. Graph the zero’s of the function
f(x) = x( x-5 ) ( x+2) ( x+ 4) ( x – 3)

11. END BEHAVIORS

12. END BEHAVIORS degree of the function leading coefficient
depend on the
degree of the function
and the sign of the
leading coefficient

13. Defn:
Degree of a Function
The largest exponent of the independent variable represents the degree of the function.
State the degree of the following polynomial functions
𝑔 𝑥 =2 𝑥 5 −4 𝑥 3 +𝑥−2
𝑓 𝑥 =5𝑥+2 𝑥 2 −6 𝑥 3 +3 5 3
𝑘 𝑥 = 4𝑥 3 +6 𝑥 11 − 𝑥 10 + 𝑥 12 1
ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥) 12 8

14. What is the leading coefficient ( a ) of each polynomial?
Defn:
Leading Coefficient ( a )
The coefficient of the term of greatest degree when the function is in standard form.
What is the leading coefficient ( a ) of each polynomial?
𝑔 𝑥 =2 𝑥 5 −4 𝑥 3 +𝑥−2
𝑓 𝑥 =5𝑥+2 𝑥 2 −6 𝑥 3 +3 8
𝑘 𝑥 = 4𝑥 3 +6 𝑥 11 − 𝑥 10 + 𝑥 12 1
ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥)

15. End Behavior of a Function
𝑓 𝑥 =𝑎 𝑥 𝑛 and n is even
𝑓 𝑥 =−𝑎 𝑥 𝑛 and n is even
𝑓 𝑥 =𝑎 𝑥 𝑛 and n is odd
𝑓 𝑥 =−𝑎 𝑥 𝑛 and n is odd

16. End Behavior
Odd degree
a > 0
a < 0
Even degree

17. End Behavior
Odd degree
a > 0
a < 0
Even degree

18. End Behavior
Odd degree
a > 0
a < 0
Even degree

19. End Behavior
Odd degree
a > 0
a < 0
Even degree

20. Turning Points
The point where a function changes directions from increasing to decreasing or from decreasing to increasing.
If a function has a degree of n, then it has at most (n – 1) turning points.

21. n =7 Determine the number of turning points for the function 7 – 1 = 6x
(x + 3)
(x - 3)
f(x)=
(x + 1)
(x -1)
(x - 2)
(x + 2)
Determine the
number
of turning points
for the function -3 -2 -1 1 2 3
n =7
7 – 1 = 6

22. Turning Points
What is the most number of turning points the following polynomial functions could have?
𝑓 𝑥 =5𝑥+2 𝑥 2 −6 𝑥 3 +3
𝑔 𝑥 =2 𝑥 5 −4 𝑥 3 +𝑥−2 4
3-1
5-1 2
𝑘 𝑥 = 4𝑥 3 +6 𝑥 11 − 𝑥 10 + 𝑥 12
ℎ 𝑥 =2 𝑥 3 (4 𝑥 5 +3𝑥)
8-1 7 11
12-1

23. MULTIPLICITY
OF ZEROS

24. y = x³(x + 2)4(x − 3)5 Multiplicity
How many REAL Roots does the function have?
12 Real Roots
y = x³(x + 2)4(x − 3)5
Multiplicity   
0 is a root of multiplicity 3,
-2 is a root of multiplicity 4,
and 3 is a root of multiplicity 5.  

25. Multiplicity of Zeros
The number of times a factor (x-r) of a function is repeated is referred to as the factors multiplicity

27. Geogebra

28. Cross
EVEN
degree
ODD
degree
Bounce
“Wiggle”

31. y = (x + 2)²(x − 1)³ Multiplicity Answer.
 −2 is a root of multiplicity 2,
and 1 is a root of multiplicity 3.  
These are the 5 roots:
−2,  −2,  1,  1,  1.

32. Identify the zeros and their multiplicity
𝑓 𝑥 = 𝑥−3 𝑥+2 3
3 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
-2 is a zero with a multiplicity of 3.
Graph wiggles at the x-axis.
𝑔 𝑥 =5 𝑥+4 𝑥−7 2
-4 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
7 is a zero with a multiplicity of 2.
Graph bounces at the x-axis.
𝑔 𝑥 = 𝑥+1 (𝑥−4) 𝑥−2 2
-1 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
4 is a zero with a multiplicity of 1.
Graph crosses the x-axis.
2 is a zero with a multiplicity of 2.
Graph bounces at the x-axis.

33. Example Find the x-intercepts and multiplicity of f(x) = 2(x+2)2(x-3)
Solution:
x=-2 is a zero of multiplicity 2 or even
x=3 is a zero of multiplicity 1 or odd

34. State and graph a possible function.
𝒈 𝒙 = 𝒙+𝟏 (𝒙−𝟒) 𝒙−𝟐 𝟐
𝟎+𝟏 (𝟎−𝟒) 𝟎−𝟐 𝟐 -1 2 4 4
f(x) +∞
f(x) +∞ +1
-16 -1 2 4 1 2 1
Y = 𝟎+𝟏 (𝟎−𝟒) 𝟎−𝟐 𝟐
( 1 ) (-4) ( -2)(-2) = -16

35. Find the zeros (x intercepts) by setting polynomial = 0 and solving.
STEP 1
Determine left and right hand behavior by looking at the
highest power on x and the sign of that term.
Let’s graph:
Step 2
Find the zeros (x intercepts) by setting
polynomial = 0 and solving.
Multiplying out, highest power would be x4
Zeros are: 0, 3, -4
Step 3
Determine multiplicity of zeros.
0 multiplicity 2 (touches) 3 multiplicity 1 (crosses) -4 multiplicity 1 (crosses)
Step 4
Find and plot y intercept by putting 0 in for x
Determine maximum number of turns in graph by subtracting 1 from the degree.
Degree is 4 so maximum number of turns is 3
Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.

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