Select The Correct Answer.This Table Represents Values Of A Cubic Polynomial Function. \[ \begin{tabular}{|c|c|} \hline X$ & Y Y Y \ \hline -2 & -12 \ \hline -1 & 0 \ \hline 0 & 6 \ \hline 1 & 7.5 \ \hline 2 & 6 \ \hline 3 & 3
Introduction
In mathematics, a cubic polynomial function is a polynomial function of degree three, which means the highest power of the variable is three. These functions are commonly represented in the form of f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants. Given a table of values representing a cubic polynomial function, we can determine the function by analyzing the pattern of the values and using algebraic methods to find the coefficients of the function.
Understanding the Table
The table provided represents values of a cubic polynomial function. The table has two columns: x and y. The x column represents the input values, and the y column represents the corresponding output values. By analyzing the table, we can see that the function has a cubic relationship between the input and output values.
| x | y |
|---|---|
| -2 | -12 |
| -1 | 0 |
| 0 | 6 |
| 1 | 7.5 |
| 2 | 6 |
| 3 | 3 |
Determining the Coefficients
To determine the cubic polynomial function, we need to find the coefficients a, b, c, and d. We can use the given table to create a system of equations and solve for the coefficients.
Let's start by assuming the cubic polynomial function is in the form of f(x) = ax^3 + bx^2 + cx + d. We can substitute the given values of x and y into the function and create a system of equations.
For x = -2 and y = -12, we have:
-12 = a(-2)^3 + b(-2)^2 + c(-2) + d
Simplifying the equation, we get:
-12 = -8a + 4b - 2c + d
For x = -1 and y = 0, we have:
0 = a(-1)^3 + b(-1)^2 + c(-1) + d
Simplifying the equation, we get:
0 = -a + b - c + d
For x = 0 and y = 6, we have:
6 = a(0)^3 + b(0)^2 + c(0) + d
Simplifying the equation, we get:
6 = d
For x = 1 and y = 7.5, we have:
7.5 = a(1)^3 + b(1)^2 + c(1) + d
Simplifying the equation, we get:
7.5 = a + b + c + 6
For x = 2 and y = 6, we have:
6 = a(2)^3 + b(2)^2 + c(2) + d
Simplifying the equation, we get:
6 = 8a + 4b + 2c + 6
For x = 3 and y = 3, we have:
3 = a(3)^3 + b(3)^2 + c(3) + d
Simplifying the equation, we get:
3 = 27a + 9b + 3c + 6
Solving the System of Equations
We now have a system of six equations with four unknowns. We can solve the system using algebraic methods or numerical methods.
Let's start by solving the first equation for d:
d = 6
Substituting d into the second equation, we get:
0 = -a + b - c + 6
Simplifying the equation, we get:
-a + b - c = -6
Substituting d into the third equation, we get:
6 = d
Substituting d into the fourth equation, we get:
7.5 = a + b + c + 6
Simplifying the equation, we get:
a + b + c = 1.5
Substituting d into the fifth equation, we get:
6 = 8a + 4b + 2c + 6
Simplifying the equation, we get:
8a + 4b + 2c = 0
Substituting d into the sixth equation, we get:
3 = 27a + 9b + 3c + 6
Simplifying the equation, we get:
27a + 9b + 3c = -3
Using Substitution Method
We can use the substitution method to solve the system of equations. Let's start by solving the first equation for a:
a = (-6 + b - c) / -1
Substituting a into the second equation, we get:
0 = (-6 + b - c) / -1 + b - c + 6
Simplifying the equation, we get:
-6 + b - c = -b + c - 6
Simplifying further, we get:
2b - 2c = -12
Dividing both sides by 2, we get:
b - c = -6
Substituting b - c = -6 into the third equation, we get:
a + (-6 + c) + c = 1.5
Simplifying the equation, we get:
a + c = 7.5
Substituting a + c = 7.5 into the fourth equation, we get:
(-6 + b - c) / -1 + b - c = 1.5
Simplifying the equation, we get:
-6 + b - c = -1.5 + b - c
Simplifying further, we get:
-6 = -1.5
This is a contradiction, so we have a dependent system of equations.
Using Elimination Method
We can use the elimination method to solve the system of equations. Let's start by subtracting the second equation from the first equation:
-12 = -8a + 4b - 2c + d - (-a + b - c + d)
Simplifying the equation, we get:
-12 = -7a + 3b - c
Subtracting the third equation from the fourth equation, we get:
7.5 = a + b + c + 6 - d
Simplifying the equation, we get:
1.5 = a + b + c
Subtracting the fifth equation from the sixth equation, we get:
3 = 27a + 9b + 3c + 6 - (8a + 4b + 2c + 6)
Simplifying the equation, we get:
-3 = 19a + 5b + c
Solving the Reduced System
We now have a reduced system of three equations with three unknowns. We can solve the system using algebraic methods or numerical methods.
Let's start by solving the first equation for a:
a = (-3 + 3b - c) / -7
Substituting a into the second equation, we get:
1.5 = (-3 + 3b - c) / -7 + b + c
Simplifying the equation, we get:
1.5 = (-3 + 3b - c) / -7 + b + c
Multiplying both sides by -7, we get:
-10.5 = -3 + 3b - c + 7b + 7c
Simplifying the equation, we get:
-10.5 = 10b + 6c
Dividing both sides by 10, we get:
-1.05 = b + 0.6c
Substituting b + 0.6c = -1.05 into the third equation, we get:
(-3 + 3b - c) / -7 = -1.05 + 0.6c
Simplifying the equation, we get:
-3 + 3b - c = 7.35 - 0.6c
Simplifying further, we get:
-3 + 3b = 7.95 - 1.6c
Finding the Coefficients
We now have a system of two equations with two unknowns. We can solve the system using algebraic methods or numerical methods.
Let's start by solving the first equation for b:
b = (7.95 - 1.6c + 3) / 3
Substituting b into the second equation, we get:
-1.05 = (7.95 - 1.6c + 3) / 3 + 0.6c
Simplifying the equation, we get:
-1.05 = (10.95 - 1.6c) / 3 + 0.6c
Multiplying both sides by 3, we get:
-3.15 = 10.95 - 1.6c + 1.8c
Simplifying the equation, we get:
-3.15 = 10.95 + 0.2c
Subtracting 10.95 from both sides, we get:
-14.05 = 0.2c
Dividing both sides by 0.2, we get:
c = -70.25
Substituting c = -70.25 into the equation b = (7.95 - 1.6c + 3) / 3, we get:
b = (7.95 - 1.6(-70.25) + 3) / 3
Q&A: Determining the Cubic Polynomial Function
Q: What is a cubic polynomial function?
A: A cubic polynomial function is a polynomial function of degree three, which means the highest power of the variable is three. These functions are commonly represented in the form of f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
Q: How do I determine the coefficients of a cubic polynomial function from a given table?
A: To determine the coefficients of a cubic polynomial function from a given table, you need to create a system of equations using the values in the table. You can then solve the system of equations using algebraic methods or numerical methods.
Q: What are the steps to determine the coefficients of a cubic polynomial function?
A: The steps to determine the coefficients of a cubic polynomial function are:
- Create a system of equations using the values in the table.
- Solve the system of equations using algebraic methods or numerical methods.
- Find the values of the coefficients a, b, c, and d.
Q: How do I create a system of equations from a given table?
A: To create a system of equations from a given table, you need to substitute the values of x and y into the cubic polynomial function f(x) = ax^3 + bx^2 + cx + d. This will give you a system of equations that you can solve to find the values of the coefficients a, b, c, and d.
Q: What are some common methods for solving a system of equations?
A: Some common methods for solving a system of equations include:
- Substitution method
- Elimination method
- Graphical method
- Numerical method
Q: What is the substitution method?
A: The substitution method is a method for solving a system of equations by substituting the value of one variable into the other equation. This method is useful when one of the equations is linear and the other equation is quadratic.
Q: What is the elimination method?
A: The elimination method is a method for solving a system of equations by eliminating one of the variables. This method is useful when the coefficients of the variables are the same in both equations.
Q: What is the graphical method?
A: The graphical method is a method for solving a system of equations by graphing the two equations on a coordinate plane. This method is useful when the system of equations has a unique solution.
Q: What is the numerical method?
A: The numerical method is a method for solving a system of equations by using numerical techniques such as the Newton-Raphson method. This method is useful when the system of equations has a complex solution.
Q: How do I determine the cubic polynomial function from a given table?
A: To determine the cubic polynomial function from a given table, you need to follow these steps:
- Create a system of equations using the values in the table.
- Solve the system of equations using algebraic methods or numerical methods.
- Find the values of the coefficients a, b, c, and d.
- Substitute the values of the coefficients into the cubic polynomial function f(x) = ax^3 + bx^2 + cx + d.
Q: What are some common applications of cubic polynomial functions?
A: Some common applications of cubic polynomial functions include:
- Modeling population growth
- Modeling the motion of objects
- Modeling the behavior of electrical circuits
- Modeling the behavior of mechanical systems
Q: How do I use cubic polynomial functions in real-world applications?
A: To use cubic polynomial functions in real-world applications, you need to follow these steps:
- Identify the problem you want to solve.
- Determine the variables involved in the problem.
- Create a system of equations using the values in the table.
- Solve the system of equations using algebraic methods or numerical methods.
- Find the values of the coefficients a, b, c, and d.
- Substitute the values of the coefficients into the cubic polynomial function f(x) = ax^3 + bx^2 + cx + d.
- Use the cubic polynomial function to model the behavior of the system.