Use The Key Features Of The Polynomial $f(x)=-9x^3+8x^2-16x+3$ To Describe Its End Behavior.A. The Left Side Continues Up, And The Right Side Continues Up.B. The Left Side Continues Down, And The Right Side Continues Down.C. The Left Side
Introduction
When analyzing a polynomial function, one of the key aspects to consider is its end behavior. This refers to the behavior of the function as x approaches positive or negative infinity. In this article, we will explore the end behavior of the polynomial function f(x) = -9x^3 + 8x^2 - 16x + 3.
Key Features of the Polynomial Function
To understand the end behavior of the polynomial function, we need to examine its key features. The given function is a cubic polynomial, which means it has a degree of 3. The general form of a cubic polynomial is f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
In this case, the polynomial function is f(x) = -9x^3 + 8x^2 - 16x + 3. We can see that the leading coefficient, a, is -9. This is an important feature of the polynomial function, as it determines the end behavior of the function.
The Leading Coefficient and End Behavior
The leading coefficient, a, determines the end behavior of the polynomial function. If a is positive, the function will approach positive infinity as x approaches positive infinity, and negative infinity as x approaches negative infinity. If a is negative, the function will approach negative infinity as x approaches positive infinity, and positive infinity as x approaches negative infinity.
In this case, the leading coefficient, a, is -9, which is a negative number. Therefore, we can conclude that the function will approach negative infinity as x approaches positive infinity, and positive infinity as x approaches negative infinity.
The Degree of the Polynomial and End Behavior
The degree of the polynomial also plays a crucial role in determining the end behavior of the function. If the degree of the polynomial is even, the function will approach a horizontal asymptote as x approaches positive or negative infinity. If the degree of the polynomial is odd, the function will approach either positive or negative infinity as x approaches positive or negative infinity.
In this case, the degree of the polynomial is 3, which is an odd number. Therefore, we can conclude that the function will approach either positive or negative infinity as x approaches positive or negative infinity.
The End Behavior of the Polynomial Function
Based on the key features of the polynomial function, we can conclude that the function will approach negative infinity as x approaches positive infinity, and positive infinity as x approaches negative infinity. This is because the leading coefficient, a, is negative, and the degree of the polynomial is odd.
Conclusion
In conclusion, the end behavior of the polynomial function f(x) = -9x^3 + 8x^2 - 16x + 3 can be determined by examining its key features. The leading coefficient, a, determines the end behavior of the function, and the degree of the polynomial also plays a crucial role in determining the end behavior of the function. By analyzing these key features, we can conclude that the function will approach negative infinity as x approaches positive infinity, and positive infinity as x approaches negative infinity.
Answer Key
A. The left side continues up, and the right side continues up. B. The left side continues down, and the right side continues down. C. The left side continues down, and the right side continues up. D. The left side continues up, and the right side continues down.
The correct answer is C. The left side continues down, and the right side continues up.
Final Answer
The final answer is C.
Introduction
In our previous article, we discussed the end behavior of a polynomial function f(x) = -9x^3 + 8x^2 - 16x + 3. We examined the key features of the polynomial function, including the leading coefficient and the degree of the polynomial, to determine its end behavior. In this article, we will answer some frequently asked questions (FAQs) about the end behavior 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. It determines whether the function approaches positive or negative infinity as x approaches positive or negative infinity.
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 examine its key features, including the leading coefficient and the degree of the polynomial. If the leading coefficient is positive, the function will approach positive infinity as x approaches positive infinity, and negative infinity as x approaches negative infinity. If the leading coefficient is negative, the function will approach negative infinity as x approaches positive infinity, and positive infinity as x approaches negative infinity.
Q: What is the significance of the leading coefficient in determining the end behavior of a polynomial function?
A: The leading coefficient determines the end behavior of a polynomial function. If the leading coefficient is positive, the function will approach positive infinity as x approaches positive infinity, and negative infinity as x approaches negative infinity. If the leading coefficient is negative, the function will approach negative infinity as x approaches positive infinity, and positive infinity as x approaches negative infinity.
Q: How does the degree of the polynomial affect the end behavior of the function?
A: The degree of the polynomial also affects the end behavior of the function. If the degree of the polynomial is even, the function will approach a horizontal asymptote as x approaches positive or negative infinity. If the degree of the polynomial is odd, the function will approach either positive or negative infinity as x approaches positive or negative infinity.
Q: Can a polynomial function have a horizontal asymptote?
A: Yes, a polynomial function can have a horizontal asymptote. If the degree of the polynomial is even, the function will approach a horizontal asymptote as x approaches positive or negative infinity.
Q: How do I determine if a polynomial function has a horizontal asymptote?
A: To determine if a polynomial function has a horizontal asymptote, you need to examine its degree. If the degree of the polynomial is even, the function will approach a horizontal asymptote as x approaches positive or negative infinity.
Q: What is the difference between a polynomial function and a rational function?
A: A polynomial function is a function that can be written in the form f(x) = ax^n + bx^(n-1) + ... + cx + d, where a, b, c, and d are constants, and n is a positive integer. A rational function is a function that can be written in the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions.
Q: Can a rational function have a horizontal asymptote?
A: Yes, a rational function can have a horizontal asymptote. If the degree of the numerator is less than or equal to the degree of the denominator, the function will approach a horizontal asymptote as x approaches positive or negative infinity.
Conclusion
In conclusion, the end behavior of a polynomial function is determined by its key features, including the leading coefficient and the degree of the polynomial. By examining these features, you can determine the end behavior of a polynomial function and answer frequently asked questions about the end behavior of a polynomial function.
Answer Key
Q1: What is the end behavior of a polynomial function? A1: The end behavior of a polynomial function refers to the behavior of the function as x approaches positive or negative infinity.
Q2: How do I determine the end behavior of a polynomial function? A2: To determine the end behavior of a polynomial function, you need to examine its key features, including the leading coefficient and the degree of the polynomial.
Q3: What is the significance of the leading coefficient in determining the end behavior of a polynomial function? A3: The leading coefficient determines the end behavior of a polynomial function.
Q4: How does the degree of the polynomial affect the end behavior of the function? A4: The degree of the polynomial also affects the end behavior of the function.
Q5: Can a polynomial function have a horizontal asymptote? A5: Yes, a polynomial function can have a horizontal asymptote.
Q6: How do I determine if a polynomial function has a horizontal asymptote? A6: To determine if a polynomial function has a horizontal asymptote, you need to examine its degree.
Q7: What is the difference between a polynomial function and a rational function? A7: A polynomial function is a function that can be written in the form f(x) = ax^n + bx^(n-1) + ... + cx + d, where a, b, c, and d are constants, and n is a positive integer. A rational function is a function that can be written in the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions.
Q8: Can a rational function have a horizontal asymptote? A8: Yes, a rational function can have a horizontal asymptote.
Final Answer
The final answer is that the end behavior of a polynomial function is determined by its key features, including the leading coefficient and the degree of the polynomial.