Identify The End Behavior And The Zeros Of Function { H $}$. ${ H(x) = -x^3 - 9x^2 + 4x + 96 }$Based On These Key Features, Which Statement Is True About The Graph Representing Function { H $}$?A. The Graph Is Negative

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Introduction

In mathematics, polynomial functions are a fundamental concept in algebra and calculus. These functions are used to model various real-world phenomena, such as population growth, chemical reactions, and electrical circuits. One of the essential aspects of polynomial functions is understanding their end behavior and zeros. In this article, we will explore the end behavior and zeros of the function h(x)=βˆ’x3βˆ’9x2+4x+96{ h(x) = -x^3 - 9x^2 + 4x + 96 } and determine which statement is true about the graph representing this function.

End Behavior of a Polynomial Function

The end behavior of a polynomial function refers to the behavior of the function as x approaches positive or negative infinity. This can be determined by looking at the degree and leading coefficient of the polynomial function. In the case of the function h(x)=βˆ’x3βˆ’9x2+4x+96{ h(x) = -x^3 - 9x^2 + 4x + 96 }, the degree is 3, and the leading coefficient is -1.

Determining the End Behavior

To determine the end behavior of the function, we need to consider the following cases:

  • If the degree of the polynomial is even, the end behavior will be the same as the leading coefficient.
  • If the degree of the polynomial is odd, the end behavior will be the opposite of the leading coefficient.

In this case, the degree of the polynomial is odd (3), and the leading coefficient is -1. Therefore, the end behavior of the function will be positive as x approaches positive infinity and negative as x approaches negative infinity.

Zeros of a Polynomial Function

The zeros of a polynomial function are the values of x that make the function equal to zero. In other words, they are the solutions to the equation h(x)=0{ h(x) = 0 }. To find the zeros of the function h(x)=βˆ’x3βˆ’9x2+4x+96{ h(x) = -x^3 - 9x^2 + 4x + 96 }, we need to solve the equation βˆ’x3βˆ’9x2+4x+96=0{ -x^3 - 9x^2 + 4x + 96 = 0 }.

Solving the Equation

To solve the equation βˆ’x3βˆ’9x2+4x+96=0{ -x^3 - 9x^2 + 4x + 96 = 0 }, we can use various methods, such as factoring, synthetic division, or numerical methods. In this case, we will use factoring to solve the equation.

Factoring the Polynomial

The polynomial βˆ’x3βˆ’9x2+4x+96{ -x^3 - 9x^2 + 4x + 96 } can be factored as follows:

βˆ’x3βˆ’9x2+4x+96=βˆ’(x+8)(x2+xβˆ’12){ -x^3 - 9x^2 + 4x + 96 = -(x + 8)(x^2 + x - 12) }

Solving the Quadratic Equation

The quadratic equation x2+xβˆ’12=0{ x^2 + x - 12 = 0 } can be solved using the quadratic formula:

x=βˆ’bΒ±b2βˆ’4ac2a{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }

In this case, a = 1, b = 1, and c = -12. Plugging these values into the quadratic formula, we get:

x=βˆ’1Β±1+482{ x = \frac{-1 \pm \sqrt{1 + 48}}{2} }

x=βˆ’1Β±492{ x = \frac{-1 \pm \sqrt{49}}{2} }

x=βˆ’1Β±72{ x = \frac{-1 \pm 7}{2} }

Therefore, the solutions to the quadratic equation are:

x=βˆ’1+72=3{ x = \frac{-1 + 7}{2} = 3 }

x=βˆ’1βˆ’72=βˆ’4{ x = \frac{-1 - 7}{2} = -4 }

Finding the Zeros of the Function

The zeros of the function h(x)=βˆ’x3βˆ’9x2+4x+96{ h(x) = -x^3 - 9x^2 + 4x + 96 } are the values of x that make the function equal to zero. In this case, the zeros of the function are:

x=βˆ’8{ x = -8 }

x=3{ x = 3 }

x=βˆ’4{ x = -4 }

Conclusion

In conclusion, the end behavior of the function h(x)=βˆ’x3βˆ’9x2+4x+96{ h(x) = -x^3 - 9x^2 + 4x + 96 } is positive as x approaches positive infinity and negative as x approaches negative infinity. The zeros of the function are x=βˆ’8{ x = -8 }, x=3{ x = 3 }, and x=βˆ’4{ x = -4 }. Therefore, the statement that is true about the graph representing this function is:

The graph is negative

This statement is true because the function h(x)=βˆ’x3βˆ’9x2+4x+96{ h(x) = -x^3 - 9x^2 + 4x + 96 } is a cubic function with a negative leading coefficient, which means that the graph of the function will be negative for all values of x.

References

Additional Resources

  • Khan Academy: Polynomial Functions
  • MIT OpenCourseWare: Calculus
  • Wolfram Alpha: Polynomial Functions
    Q&A: Understanding the End Behavior and Zeros of a Polynomial Function ====================================================================

Frequently Asked Questions

In this article, we will address some of the most common questions related to the end behavior and zeros of a polynomial function.

Q: What is the end behavior of a polynomial function?

A: The end behavior of a polynomial function refers to the behavior of the function as x approaches positive or negative infinity. This can be determined by looking at the degree and leading coefficient of the polynomial function.

Q: How do I determine the end behavior of a polynomial function?

A: To determine the end behavior of a polynomial function, you need to consider the following cases:

  • If the degree of the polynomial is even, the end behavior will be the same as the leading coefficient.
  • If the degree of the polynomial is odd, the end behavior will be the opposite of the leading coefficient.

Q: What are the zeros of a polynomial function?

A: The zeros of a polynomial function are the values of x that make the function equal to zero. In other words, they are the solutions to the equation h(x)=0{ h(x) = 0 }.

Q: How do I find the zeros of a polynomial function?

A: To find the zeros of a polynomial function, you can use various methods, such as factoring, synthetic division, or numerical methods. In this case, we used factoring to solve the equation.

Q: What is the significance of the zeros of a polynomial function?

A: The zeros of a polynomial function are significant because they represent the points where the function intersects the x-axis. This can be useful in various applications, such as graphing and solving equations.

Q: Can you provide an example of a polynomial function with a negative leading coefficient?

A: Yes, an example of a polynomial function with a negative leading coefficient is h(x)=βˆ’x3βˆ’9x2+4x+96{ h(x) = -x^3 - 9x^2 + 4x + 96 }. This function has a negative leading coefficient, which means that the graph of the function will be negative for all values of x.

Q: How do I graph a polynomial function?

A: To graph a polynomial function, you can use various methods, such as plotting points, using a graphing calculator, or using a computer algebra system. In this case, we used a graphing calculator to visualize the graph of the function.

Q: What are some common applications of polynomial functions?

A: Polynomial functions have numerous applications in various fields, such as:

  • Physics: Polynomial functions are used to model the motion of objects, including the trajectory of projectiles and the vibration of springs.
  • Engineering: Polynomial functions are used to design and analyze electrical circuits, mechanical systems, and other engineering applications.
  • Economics: Polynomial functions are used to model economic systems, including the behavior of supply and demand curves.

Q: Can you provide some additional resources for learning about polynomial functions?

A: Yes, some additional resources for learning about polynomial functions include:

  • Khan Academy: Polynomial Functions
  • MIT OpenCourseWare: Calculus
  • Wolfram Alpha: Polynomial Functions

Conclusion

In conclusion, understanding the end behavior and zeros of a polynomial function is essential for various applications in mathematics and other fields. By following the steps outlined in this article, you can determine the end behavior and zeros of a polynomial function and visualize its graph using a graphing calculator or computer algebra system.

References

Additional Resources

  • Khan Academy: Polynomial Functions
  • MIT OpenCourseWare: Calculus
  • Wolfram Alpha: Polynomial Functions