2.3 Polynomial Functions
The objectives of this lesson will help you learn how to:
1) Graph polynomial functions.
2) Predict their end behavior.
3) Determine number of extrema and zeros.
4) Understand multiplicity.
5) Observe the x-intercepts of odd and even power.
6) Investigate calculator error in viewing window.
1. Investigate graphs of polynomial functions with odd and even degree. (Flatter takes on the
next odd or even power)
Y = x,
Y = 𝑥3,
Y = 𝑥5
Y = 𝑥7
Y = 𝑥2
Y = 𝑥4
Y = 𝑥6
Y = 𝑥8
2. Predict the end behavior and write it using limit notation. (Same direction for even so has
absolute or global max/min. Opposite direction for odd so no absolute max/min.)
3. Determine the number of local extrema and zeroes. (# of zeros = number of degree
including real zeros and imaginary zeros, # of extrema = n – 1 where n = the degree.) Show a
graph with 1 extrema to predict number of degree = 2. Show graph of 2 extrema to predict # of
degree = 3. Show graph of 5 extrema to predict # of degree = 6)
Function
Y=x
# of
# of
degree at
most
zeros
1
1
𝑌1 = .5(x+3)(x -2)(x-5)
3
3
5
5
7
7
𝒀𝟑 = (x+8)(x-9) 𝒀𝟐
Scale -100000, 50000, 10000
Predict: Leading coefficient positive
11
11
10
None
Predict: Leading coefficient negative
13
13
12
None
Scale -10,20,1
𝑌2 = -.5(x+3)(x -2)(x-5)(x+5)(x -7) or
𝒀𝟐 = -1(x+5)(x -7) 𝒀𝟏
Scale -400, 600, 50
𝑌3 = -.5(x+3)(x -2)(x-5)(x+5)(x 7)(x+8)(x-9) or
# of at
Absolute End Behavior
most
Max/Min
local
extrema
0
None
X →∞, y→∞
X →-∞, y→-∞
Opposite
direction
2
None
X →∞, y→∞
X →-∞, y→-∞
Opposite
direction
4
None
X →∞, y→-∞
X →-∞, y→∞
Opposite
direction
6
None
X →∞, y→-∞
X →-∞, y→∞
Opposite
direction
X →∞, y→∞
X →-∞, y→-∞
Opposite
direction
X →∞, y→-∞
X →-∞, y→∞
Opposite
direction
X →∞, y→∞
X →-∞, y→∞
Same direction
X →∞, y→∞
X →-∞, y→∞
Same direction
𝑌1 = .5(x+3)(x -2)
2
2
1
Absolute
Min
𝑌2 = .5(x+3)(x -2)(x-5)(x+5) or
4
4
3
Absolute
Min
6
6
5
Absolute
Max
X →∞, y→-∞
X →-∞, y→-∞
Same direction
𝒀𝟑 = -1(x+8)(x – 9) 𝒀𝟐
Scale -10000,30000,5000
Predict: Leading coefficient positive
10
10
9
Absolute
Min
Predict: Leading coefficient negative
12
12
11
Absolute
Max
X →∞, y→∞
X →-∞, y→∞
Same direction
X →∞, y→-∞
X →-∞, y→-∞
Same direction
𝒀𝟐 = (x-5)(x+5) 𝒀𝟏
Scale -100,100,10
𝒀𝟑 = -.5(x+3)(x -2)(x-5) (x+5)(x+8)(x
-9) or
4. Multiplicity of zeros or x-intercepts.
a)
Y = (𝑥 − 3)2 = (𝑥 − 3)(𝑥 − 3) . X = 3 has two zeros. At x = 3 there is a multiplicity of
2. F(3) = 0 has multiplicity 2. Observe graph.
b)
Y = (𝑥 + 4)(𝑥 + 2)3 (𝑥 − 1)2 = (𝑥 + 4)(𝑥 + 2)(𝑥 + 2)(𝑥 + 2)(𝑥 − 1)(𝑥 − 1).
Multiplicity 1 at x = -4. Multiplicity 3 at x = -2. Multiplicity 2 at x =1. Observe graph.
5.
Zeros of odd and even multiplicity.
a) If the function crosses x-axis in a straight line then the degree of that linear factor is odd such
as 1. If it crosses flatter then the degree takes on the next odd such as 3, and flatter even more
than the degree becomes 5, and so on.
b) If the function does not crosses the x-axis but goes back in the direction it came from then the
degree of that linear function is even, such as 2. If it does not cross but becomes flatter then the
degree takes on the next even power such as 4, and flatter even more than they degree becomes
6, and so on.
Ex.1 Y =(𝑥 − 4)(𝑥 + 2). The graph crosses the x-axis at x = -2 in a straight line. So it takes on
the first odd power. The graph also has x-intercept at x = 4. It crosses in a straight line so it takes
on the first odd power. Degree 1 + Degree 1 = 2nd degree or a squaring function.
Ex.2 y = (𝑥 + 3)(𝑥 + 3)(𝑥 − 4). The graph does not cross the x-axis at x = -3 since there the
multiplicity 2 is even. At x = -3 the graph goes back in the direction where it came from.
The graph crosses the x-axis at x = 4 since the multiplicity 1 is odd.
6.
Sketch a possible graph for the following function without the use of calculator.
a) The leading coefficient is positive. There is a multiplicity of 3 at x = -4. There is a multiplicity
of 2 at x = 5.
b) The leading coefficient is negative. There is a multiplicity of 1 at x = -7. There is a
multiplicity of 2 at x = -4. There is a multiplicity of 3 at x = 2. There is a multiplicity
of 4 at x = 6.
7.
How to determine whether the calculator is displaying the entire graph?
The function to be graphed is y = 3𝑥 4 + 2𝑥 3 – 7𝑥 2 + 2x − 3. Graph using scale x at -1, 2, .5 and
y at -10, 10, 1. What must be true of the graph?
a) It must show at most 3 extrema.
b) It must show at most 4 roots.
c) Since the leading coefficient is positive and the power is even the graph must show that the
end behavior is going in the same direction toward positive infinity. As X →∞, y→∞ and as X
→-∞, y→∞
The function to be graphed is y = −9𝑥 3 +27𝑥 2 + 54x − 73. Graph using scale x at -3, 1.5,.5 and
y at -100, 100, 10. What must be true of the graph?
a) It must show at most 2 extrema.
b) It must show at most 3 roots.
c) Since the leading coefficient is negative and the power is odd the graph must show that the
end behavior is going in the opposite direction. As X →∞, y→-∞ and as X →-∞, y→∞