Given The Function \$g(x)=x^3-9x^2+2x+6$[/tex\], Determine The Following:- Degree:- Leading Coefficient:- End Behavior:

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The given function is a cubic polynomial, denoted as g(x) = x^3 - 9x^2 + 2x + 6. In this article, we will determine the degree, leading coefficient, and end behavior of the function g(x).

Degree of the Function

The degree of a polynomial is the highest power of the variable (in this case, x) in the polynomial. To determine the degree of the function g(x), we need to identify the term with the highest power of x.

g(x) = x^3 - 9x^2 + 2x + 6

In the above expression, the term with the highest power of x is x^3. Therefore, the degree of the function g(x) is 3.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable. In the given function g(x), the leading coefficient is the coefficient of the term x^3.

g(x) = x^3 - 9x^2 + 2x + 6

The coefficient of the term x^3 is 1. Therefore, the leading coefficient of the function g(x) is 1.

End Behavior

The end behavior of a function refers to the behavior of the function as x approaches positive infinity or negative infinity. To determine the end behavior of the function g(x), we need to analyze the degree and leading coefficient of the function.

Since the degree of the function g(x) is 3, which is an odd number, the end behavior of the function will be different for positive and negative values of x.

For x > 0, the function g(x) will approach positive infinity as x approaches positive infinity.

For x < 0, the function g(x) will approach negative infinity as x approaches negative infinity.

Graphical Representation

To visualize the end behavior of the function g(x), we can plot the function on a graph.

g(x) = x^3 - 9x^2 + 2x + 6

The graph of the function g(x) will have a positive x-intercept and a negative y-intercept.

Conclusion

In conclusion, the degree of the function g(x) = x^3 - 9x^2 + 2x + 6 is 3, the leading coefficient is 1, and the end behavior of the function is different for positive and negative values of x.

Key Takeaways

  • The degree of a polynomial is the highest power of the variable in the polynomial.
  • The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable.
  • The end behavior of a function refers to the behavior of the function as x approaches positive infinity or negative infinity.

References

Further Reading

In the previous article, we discussed the degree, leading coefficient, and end behavior of the function g(x) = x^3 - 9x^2 + 2x + 6. In this article, we will answer some frequently asked questions related to the function g(x).

Q: What is the degree of the function g(x)?

A: The degree of the function g(x) is 3, which is the highest power of the variable (x) in the polynomial.

Q: What is the leading coefficient of the function g(x)?

A: The leading coefficient of the function g(x) is 1, which is the coefficient of the term x^3.

Q: What is the end behavior of the function g(x)?

A: The end behavior of the function g(x) is different for positive and negative values of x. For x > 0, the function g(x) will approach positive infinity as x approaches positive infinity. For x < 0, the function g(x) will approach negative infinity as x approaches negative infinity.

Q: How can I determine the degree of a polynomial?

A: To determine the degree of a polynomial, you need to identify the term with the highest power of the variable. In the case of the function g(x) = x^3 - 9x^2 + 2x + 6, the term with the highest power of x is x^3, which has a degree of 3.

Q: How can I determine the leading coefficient of a polynomial?

A: To determine the leading coefficient of a polynomial, you need to identify the coefficient of the term with the highest power of the variable. In the case of the function g(x) = x^3 - 9x^2 + 2x + 6, the coefficient of the term x^3 is 1, which is the leading coefficient.

Q: What is the significance of the degree and leading coefficient of a polynomial?

A: The degree and leading coefficient of a polynomial are important because they determine the end behavior of the function. A polynomial with an odd degree will have a different end behavior for positive and negative values of x, while a polynomial with an even degree will have the same end behavior for positive and negative values of x.

Q: How can I graph the function g(x) = x^3 - 9x^2 + 2x + 6?

A: To graph the function g(x) = x^3 - 9x^2 + 2x + 6, you can use a graphing calculator or a computer algebra system. You can also plot the function on a graph by hand using a table of values.

Q: What are some real-world applications of the function g(x) = x^3 - 9x^2 + 2x + 6?

A: The function g(x) = x^3 - 9x^2 + 2x + 6 has many real-world applications, including modeling population growth, predicting the behavior of physical systems, and optimizing functions.

Q: How can I use the function g(x) = x^3 - 9x^2 + 2x + 6 to solve problems?

A: You can use the function g(x) = x^3 - 9x^2 + 2x + 6 to solve problems by substituting values of x into the function and evaluating the resulting expression. You can also use the function to model real-world situations and make predictions about the behavior of physical systems.

Conclusion

In conclusion, the function g(x) = x^3 - 9x^2 + 2x + 6 is a cubic polynomial with a degree of 3 and a leading coefficient of 1. The end behavior of the function is different for positive and negative values of x. We hope that this Q&A article has helped you to understand the function g(x) and its applications.

Key Takeaways

  • The degree of a polynomial is the highest power of the variable in the polynomial.
  • The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable.
  • The end behavior of a function refers to the behavior of the function as x approaches positive infinity or negative infinity.
  • The function g(x) = x^3 - 9x^2 + 2x + 6 has many real-world applications, including modeling population growth, predicting the behavior of physical systems, and optimizing functions.

References

Further Reading