1.6 Polynomial functions and end behavior
Consider the polynomial function f(x) = 2x^3 - 5x^2 + 3x - 1.
Complete the table by calculating the missing values of f(x).
| x | -1000 | -100 | -10 | 0 | 10 | 100 | 1000 |
|---|---|---|---|---|---|---|---|
| f(x) | -1 |
What happens to values of f(x) as x-values increases?
What happens to values of f(x) as x-values decreases?
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?
Given the following polynomial functions, determine the degree and leading term of each function.
Define the term "end behavior" in the context of polynomial functions.
What is the significance of the degree of a polynomial function in determining its end behavior?
How do even and odd degrees in polynomial functions affect their end behavior?
Given the following information about a polynomial function, determine the end behavior of the function using limit notation.
The polynomial function has a degree of 5 and a positive leading coefficient.
The polynomial function has an even degree and a negative leading coefficient.
The polynomial function has an odd degree and a negative leading coefficient.
The polynomial function has an even degree and a positive leading coefficient.
Describe the end behavior of each function by completing the following statement:
As ⬚ approaches positive infinity, ⬚ approaches ⬚; as ⬚ approaches negative infinity, ⬚ approaches ⬚.
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.
f\left(x\right) = -2x^3 + 5x^2 - 7x + 3
f(x) = 3x^4 - 5x^2 + 2x - 7
g(x) = -4x^5 + 3x^3 - 8x + 12
g(x) = 2x^3 - 7x^2 + 4x + 1
h(x) = 2(x - 3)^2(x + 1)(x - 1)^3
h(x) = -3(x + 2)(x - 4)^2(x + 5)^3
For the following graphs:
Describe the end behavior of the polynomial function as x \to -\infty and x \to \infty using limit notation.
State whether the degree is odd or even.
State whether the leading coefficient is positive or negative.
The polynomial function g\left(x\right) = 4x^5 - 3x^2 + 8x - 6 models the growth of a population over time.
Describe the end behavior of the function using limit notation and the rise and fall of its graph.
Explain the meaning of the end behavior in the context of population growth.
At what interval will the model be realistic?
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.
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.
Determine the end behavior of the function as time increases and decreases without bound using limit notation.
Interpret the end behavior in terms of the virus growth.
For the polynomial function p\left(x\right) = -5x^7 + 6x^5 - x^3 + 2x + 1.
Use limit notation to describe the end behavior as x \to \pm \infty.
Explain what the end behavior reveals about the long-term behavior of the function.
Consider the polynomial function f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 4.
Analyze the relationship between the values of the leading term and lower-degree terms as the input values increase without bound.
Describe how the lower-degree terms affect the function when x \to +\infty.
Suppose a polynomial function g(x) = ax^3 + bx^2 + cx + d has a degree of 3 and a positive leading coefficient.
Describe the relationship between the values of the leading term and lower-degree terms as the input values decrease without bound.
What happens to the function when x \to -\infty?
Consider the polynomial function P(x) = 3x^5 - 4x^4 + 2x^2 - x + 1.
Describe the end behavior of P(x).
Determine a new possible polynomial function if the end behavior is reversed. Justify your answer.
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.
Analyze the end behavior of the function.
Explain what the end behavior suggests about the company's long-term profitability.
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.
Determine the end behavior of the function.
Explain what the end behavior suggests about the projectile's trajectory over time.
Determine at what time will the projectile fall to the ground.
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.
Analyze the end behavior of the function.
Discuss the end behavior's implications for the long-term temperature stability in the room.
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.
Estimate the end behavior of the polynomial function by analyzing its graph.
Explain the implications of this end behavior on the long-term population trends of the species in the wildlife reserve.