Which polynomial(s) could have the following end behavior: as x→∞, f(x)→∞, and as x→−∞, f(x)→∞?
There may be more than one correct answer. Select all correct answers.
ax
−ax^5+bx^2+cx
ax^6
ax^2−b
ax^4−bx^3−cx^2−dx+e
ax^3+bx^2+c
this was so briefly went over in my notes...just wondering if I can get input how to know which is related to the end behavior formula given
any polynomial of even degree, with positive leading coefficient, will exhibit the required behavior.
Think of x^2. It goes up on both ends. Higher even degree will too, possibly with some wiggles near where it crosses the x-axis.
To determine which polynomial(s) could have the given end behavior, you can consider the degree and leading coefficient of the polynomial. Here's a step-by-step approach to solve this problem:
1. Recall that the degree of a polynomial is determined by the highest power of x with a non-zero coefficient. For example, the degree of the polynomial ax^2 + bx + c is 2.
2. As x approaches infinity or negative infinity, the behavior of a polynomial is primarily determined by the term with the highest degree.
3. If the leading coefficient (the coefficient of the term with the highest power of x) is positive, as x approaches infinity, the polynomial also approaches positive infinity.
4. Conversely, if the leading coefficient is negative, as x approaches infinity, the polynomial approaches negative infinity.
Now, let's apply these steps to the given options:
1. ax: This is a linear polynomial with a degree of 1. As x approaches infinity, the behavior depends on the sign of the leading coefficient. If a is positive, the polynomial approaches positive infinity. If a is negative, the polynomial approaches negative infinity. Therefore, this option satisfies the given end behavior.
2. -ax^5 + bx^2 + cx: This is a polynomial of degree 5. As x approaches infinity, the behavior is primarily determined by the first term (-ax^5). If a is negative, the polynomial approaches negative infinity. Hence, this option satisfies the given end behavior.
3. ax^6: This is a polynomial of degree 6. As x approaches infinity, the behavior is determined by the first term (ax^6). If a is positive, the polynomial approaches positive infinity. Thus, this option also satisfies the given end behavior.
4. ax^2 - b: This is a quadratic polynomial with a degree of 2. As x approaches infinity, the behavior is determined by the first term (ax^2). If a is positive, the polynomial approaches positive infinity, satisfying the given end behavior.
5. ax^4 - bx^3 - cx^2 - dx + e: This polynomial has a degree of 4. As x approaches infinity, the behavior is determined by the first term (ax^4). If a is positive, the polynomial approaches positive infinity, satisfying the given end behavior.
6. ax^3 + bx^2 + c: This is a polynomial of degree 3. As x approaches infinity, the behavior is determined by the term with the highest degree (ax^3). If a is positive, the polynomial approaches positive infinity, satisfying the given end behavior.
Based on the analysis above, the polynomials that could have the given end behavior are:
- ax
- -ax^5 + bx^2 + cx
- ax^6
- ax^2 - b
- ax^4 - bx^3 - cx^2 - dx + e
- ax^3 + bx^2 + c
Please note that there might be other correct answers as well, depending on the coefficients a, b, c, d, e.
To determine the end behavior of a polynomial, we need to consider the degree and leading coefficient of the polynomial.
The end behavior of a polynomial is determined by its highest degree term. We consider the sign of the leading coefficient and the parity (even or odd) of the degree.
1. If the leading coefficient is positive and the degree is even, then as x approaches positive infinity (+∞), the function will also approach positive infinity, and as x approaches negative infinity (-∞), the function will also approach positive infinity.
2. If the leading coefficient is positive and the degree is odd, then as x approaches positive infinity (+∞), the function will approach positive infinity, and as x approaches negative infinity (-∞), the function will approach negative infinity.
3. If the leading coefficient is negative and the degree is even, then as x approaches positive infinity (+∞), the function will approach negative infinity, and as x approaches negative infinity (-∞), the function will approach positive infinity.
4. If the leading coefficient is negative and the degree is odd, then as x approaches positive infinity (+∞), the function will approach negative infinity, and as x approaches negative infinity (-∞), the function will also approach negative infinity.
Let's analyze each polynomial in the given options:
1. ax: Since the degree is 1, it is odd. The leading coefficient can be either positive or negative. So, depending on the value of a, it can have end behavior as x→∞, f(x)→∞, and as x→−∞, f(x)→∞. This polynomial could be a correct answer.
2. −ax^5+bx^2+cx: The degree is 5, which is odd. The leading term is −ax^5. As x approaches positive infinity (+∞), this term becomes negative infinity, and as x approaches negative infinity (-∞), this term becomes positive infinity. Therefore, this polynomial does not have the desired end behavior.
3. ax^6: Since the degree is 6, it is even. The leading coefficient can be either positive or negative. So, depending on the value of a, it can have end behavior as x→∞, f(x)→∞, and as x→−∞, f(x)→∞. This polynomial could be a correct answer.
4. ax^2−b: Since the degree is 2, it is even. The leading coefficient can be either positive or negative. So, depending on the value of a, it can have end behavior as x→∞, f(x)→∞, and as x→−∞, f(x)→∞. This polynomial could be a correct answer.
5. ax^4−bx^3−cx^2−dx+e: Since the degree is 4, it is even. The leading term is ax^4. As x approaches positive infinity (+∞), this term becomes positive infinity, and as x approaches negative infinity (-∞), this term becomes positive infinity. Therefore, this polynomial does not have the desired end behavior.
6. ax^3+bx^2+c: The degree is 3, which is odd. The leading term is ax^3. As x approaches positive infinity (+∞), this term becomes positive infinity, and as x approaches negative infinity (-∞), this term becomes negative infinity. Therefore, this polynomial does not have the desired end behavior.
Based on the explanations above, the polynomials that could have the desired end behavior are:
- ax
- ax^6
- ax^2−b
Hope this helps!