R(x) = −x5 + 4x3 − 3x
(a) Describe the end behavior of the polynomial function.
End behavior: y → rises to the right
Incorrect: Your answer is incorrect.
as x → ∞
y → falls to the left
Incorrect: Your answer is incorrect.
as x → −∞
can someone explain what I did wrong here? and make corrections if I did?
You have correctly identified that the end behavior of the polynomial function R(x) can be described as "rises to the right" as x approaches positive infinity. However, your statement for the end behavior as x approaches negative infinity is incorrect.
The correct end behavior as x approaches negative infinity can be described as "falls to the left." This means that the values of the function decrease without bound as x becomes increasingly negative.
So the corrected description of the end behavior of the polynomial function R(x) is:
- As x approaches positive infinity, y rises to the right.
- As x approaches negative infinity, y falls to the left.
In order to determine the end behavior of a polynomial function, you need to look at the leading term of the function, which is the term with the highest degree. In this case, the leading term is -x^5.
When x approaches infinity (x → ∞), the sign of the leading term remains negative because raising positive numbers to odd powers results in positive numbers, and multiplying them by -1 results in negative numbers. So, as x becomes larger and larger, the value of the function, R(x), will also become more and more negative. Therefore, the correct description of the end behavior is: y → falls to the left as x → ∞.
When x approaches negative infinity (x → -∞), the sign of the leading term remains negative because raising negative numbers to odd powers also results in negative numbers. So, as x becomes more and more negative, the value of the function, R(x), will also become more and more negative. Therefore, the correct description of the end behavior is: y → falls to the left as x → -∞.
Therefore, the correct answers are:
- as x → ∞, y → falls to the left.
- as x → -∞, y → falls to the left.
as x→∞, -x^5 becomes more and more negative: it falls to the right: y→-∞
Similarly, as x→-∞, -x^5 rises to the left: y→∞
Since x^5 becomes so much greater than any lower powers of x, only the highest power needs to be considered.
I think maybe you missed the leading minus sign on x^5