describe the end behavior of the function f(x)=-2x^4-x^3+3
The end behavior of a function describes what happens to the function as x approaches either positive infinity or negative infinity.
In the function f(x) = -2x^4 - x^3 + 3, the leading term is -2x^4. Since the exponent of x is even (4 is an even number), the end behavior of the function will behave similarly to the end behavior of the leading term.
As x approaches positive infinity, -2x^4 becomes larger and larger, and since the coefficient is negative (-2), the function will approach negative infinity. This means that the function will decrease without bound as x gets larger and larger.
As x approaches negative infinity, -2x^4 also becomes larger and larger, but since the coefficient is negative (-2), the function will again approach negative infinity. This means that the function will decrease without bound as x gets smaller and smaller.
In summary, as x approaches positive or negative infinity, the function f(x) = -2x^4 - x^3 + 3 will approach negative infinity.
To determine the end behavior of the function f(x) = -2x^4 - x^3 + 3, you need to observe what happens to the function as x approaches positive and negative infinity.
1. Look at the leading term: The leading term is -2x^4. The degree of the polynomial function is 4, and the leading coefficient is negative.
2. Consider the evenness of the degree: Since the degree of the function is even, it suggests that the end behavior will be similar on both sides of the y-axis.
Now, let's break down the end behavior in the following steps:
As x approaches positive infinity:
- Since the degree of the polynomial is even and the leading coefficient is negative, the function will head towards negative infinity as x becomes very large. This indicates that the graph will dip downward towards the negative y-axis.
As x approaches negative infinity:
- Similar to the positive infinity case, as x becomes increasingly negative, the function will also go towards negative infinity. The graph will rise upward, moving from the third quadrant to the second quadrant.
In summary, the end behavior of the function f(x) = -2x^4 - x^3 + 3 is as follows:
- As x approaches positive infinity, y approaches negative infinity.
- As x approaches negative infinity, y also approaches negative infinity.
Note: This analysis holds true for all quadratic functions with an even degree and a negative leading coefficient.
To describe the end behavior of the function f(x) = -2x^4 - x^3 + 3, we need to analyze what happens as x approaches positive infinity and negative infinity.
As x approaches positive infinity (x → ∞), the term with the highest exponent dominates the function. In this case, -2x^4 has the largest exponent. When x is very large, the negative sign in front of -2x^4 will cause the function to approach negative infinity.
So, as x approaches positive infinity, f(x) approaches negative infinity (f(x) → -∞).
When x approaches negative infinity (x → -∞), the same reasoning applies. The term with the highest exponent, -2x^4, dominates the function. As x becomes very large and negative, the negative sign in front of -2x^4 will cause the function to approach positive infinity.
Therefore, as x approaches negative infinity, f(x) approaches positive infinity (f(x) → ∞).
In summary, the end behavior of the function f(x) = -2x^4 - x^3 + 3 is as follows:
- As x approaches positive infinity, f(x) approaches negative infinity (f(x) → -∞).
- As x approaches negative infinity, f(x) approaches positive infinity (f(x) → ∞).