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1.6 Polynomial functions and end behavior

Worksheet
What do you remember?
1

Consider the polynomial function f(x) = 2x^3 - 5x^2 + 3x - 1.

a

Complete the table by calculating the missing values of f(x).

x-1000-100-100101001000
f(x)-1
b

What happens to values of f(x) as x-values increases?

c

What happens to values of f(x) as x-values decreases?

d

What would be the changes in the values of f(x) when x-values increases or decreases if the function is changed to f(x) = -2x^3 - 5x^2 + 3x - 1?

2

Given the following polynomial functions, determine the degree and leading term of each function.

a
f(x) = -x^4 - 2x^2 + x - 7
b
g(x) = 2(x - 1)(x + 1)(x - 3)
c
h(x) = \dfrac 15 x^5 - 5x^2 + 3x - 1
d
k(x) = 6 - 4x^2 + x^4
3

Define the term "end behavior" in the context of polynomial functions.

4

What is the significance of the degree of a polynomial function in determining its end behavior?

5

How do even and odd degrees in polynomial functions affect their end behavior?

6

Given the following information about a polynomial function, determine the end behavior of the function using limit notation.

a

The polynomial function has a degree of 5 and a positive leading coefficient.

b

The polynomial function has an even degree and a negative leading coefficient.

c

The polynomial function has an odd degree and a negative leading coefficient.

d

The polynomial function has an even degree and a positive leading coefficient.

7

Describe the end behavior of each function by completing the following statement:

As ⬚ approaches positive infinity, ⬚ approaches ⬚; as ⬚ approaches negative infinity, ⬚ approaches ⬚.

a
f(x) = 4x^2 - 3x + 1
b
C(t) = -6t^3 + 2t^2 - t + 5
c
h(x) = -x^4 + 5x^3 - 2x^2 + x - 4
d
k(t) = x^5 + 3x^3 - 4x + 2
Let's practice
8

Predict the end behavior of the following functions as x \to \pm \infty using limit notation and the rise and fall of the graph of the functions.

a

f\left(x\right) = -2x^3 + 5x^2 - 7x + 3

b

f(x) = 3x^4 - 5x^2 + 2x - 7

c

g(x) = -4x^5 + 3x^3 - 8x + 12

d

g(x) = 2x^3 - 7x^2 + 4x + 1

e

h(x) = 2(x - 3)^2(x + 1)(x - 1)^3

f

h(x) = -3(x + 2)(x - 4)^2(x + 5)^3

9

For the following graphs:

i

Describe the end behavior of the polynomial function as x \to -\infty and x \to \infty using limit notation.

ii

State whether the degree is odd or even.

iii

State whether the leading coefficient is positive or negative.

a
-9
-6
-3
3
6
9
x
-90
-60
-30
30
60
90
y
b
-8
-6
-4
-2
2
4
6
8
x
-600
-400
-200
200
400
600
y
c
-8
-6
-4
-2
2
4
x
-90
-80
-70
-60
-50
-40
-30
-20
-10
10
20
30
40
y
d
-8
-6
-4
-2
2
4
6
8
x
-40
-30
-20
-10
10
20
30
40
y
10

The polynomial function g\left(x\right) = 4x^5 - 3x^2 + 8x - 6 models the growth of a population over time.

a

Describe the end behavior of the function using limit notation and the rise and fall of its graph.

b

Explain the meaning of the end behavior in the context of population growth.

c

At what interval will the model be realistic?

11

Analyze the end behavior of the polynomial function h\left(x\right) = x^4 - 2x^3 + x^2 - 3 using limit notation. Explain the reasoning behind your answer.

12

A scientist is studying the spread of a virus and models its growth with the polynomial function v\left(x\right) = 3x^6 - 7x^4 + 2x^2 - 5, wherex represents time in days.

a

Determine the end behavior of the function as time increases and decreases without bound using limit notation.

b

Interpret the end behavior in terms of the virus growth.

13

For the polynomial function p\left(x\right) = -5x^7 + 6x^5 - x^3 + 2x + 1.

a

Use limit notation to describe the end behavior as x \to \pm \infty.

b

Explain what the end behavior reveals about the long-term behavior of the function.

14

Consider the polynomial function f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 4.

a

Analyze the relationship between the values of the leading term and lower-degree terms as the input values increase without bound.

b

Describe how the lower-degree terms affect the function when x \to +\infty.

15

Suppose a polynomial function g(x) = ax^3 + bx^2 + cx + d has a degree of 3 and a positive leading coefficient.

a

Describe the relationship between the values of the leading term and lower-degree terms as the input values decrease without bound.

b

What happens to the function when x \to -\infty?

Let's extend our thinking
16

Consider the polynomial function P(x) = 3x^5 - 4x^4 + 2x^2 - x + 1.

a

Describe the end behavior of P(x).

b

Determine a new possible polynomial function if the end behavior is reversed. Justify your answer.

17

A company's profit over time can be modeled by the polynomial function P\left(t\right) = -2t^3 + 5t^2 + 20t - 10, where t is the time in years since the company started its operations.

a

Analyze the end behavior of the function.

b

Explain what the end behavior suggests about the company's long-term profitability.

18

The height of a projectile launched from the ground can be modeled by the polynomial function h\left(t\right) = -5t^2 + 20t, where t is the time in seconds since the projectile was launched.

a

Determine the end behavior of the function.

b

Explain what the end behavior suggests about the projectile's trajectory over time.

c

Determine at what time will the projectile fall to the ground.

19

The temperature in a room as a function of time can be modeled by the polynomial function T\left(t\right) = 2t^4 - 12t^3 + 22t^2 - 15t + 70, where t is the time in hours since the heating system was turned on.

a

Analyze the end behavior of the function.

b

Discuss the end behavior's implications for the long-term temperature stability in the room.

20

The graph below shows the polynomial function P(x), which models the population of a certain species of animal in a wildlife reserve over time, where x is the number of years since the start of observation, and P(x) is the population in thousands.

2
4
6
8
10
12
14
16
18
\text{Years}
-50
50
100
150
200
\text{Population (thousands)}
a

Estimate the end behavior of the polynomial function by analyzing its graph.

b

Explain the implications of this end behavior on the long-term population trends of the species in the wildlife reserve.

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Outcomes

1.6.A

Describe end behaviors of polynomial functions.

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