Describe the end behavior of f(x)=2x^4-5x+1:
A. x-->�‡, f(x)-->�‡; x-->-�‡, f(x)-->-�‡
B. x-->�‡, f(x)-->�‡; x-->-�‡, f(x)-->�‡
C. x-->�‡, f(x)-->-�‡; x-->-�‡, f(x)-->�‡
D. x-->�‡, f(x)-->-�‡; x-->-�‡, f(x)-->-�‡
To determine the end behavior of the function f(x)=2x^4-5x+1, we need to look at the leading term, which is 2x^4.
The degree of the polynomial is 4, which means the highest power of x in the function is 4.
When the degree of a polynomial is even and the leading coefficient is positive (in this case, 2 is positive), the end behavior is as follows:
- As x approaches positive infinity, f(x) approaches positive infinity.
- As x approaches negative infinity, f(x) approaches positive infinity.
Therefore, the correct answer is option A: x-->�‡, f(x)-->�‡; x-->-�‡, f(x)-->-�‡.
To determine the end behavior of a function, we need to analyze the behavior of the function as x approaches positive infinity and negative infinity.
In the given function f(x) = 2x^4 - 5x + 1, we can see that the highest power of x is 4. This means that the function is a quartic function.
For quartic functions, the end behavior can be determined by the leading term. The leading term is the term with the highest power of x, in this case, 2x^4.
When the leading term is positive (coefficient of x^4 is positive), the end behavior as x approaches positive infinity and negative infinity will be the same. The function will either increase or decrease without bound.
When the leading term is negative (coefficient of x^4 is negative), the end behavior as x approaches positive infinity and negative infinity will be different. The function will behave oppositely as x approaches positive infinity compared to as x approaches negative infinity.
In the given function f(x) = 2x^4 - 5x + 1, the coefficient of x^4 is positive (2 is positive). Therefore, the end behavior is as follows:
- As x approaches positive infinity, f(x) approaches positive infinity.
- As x approaches negative infinity, f(x) also approaches positive infinity.
So, the correct answer is A. x--> ‡, f(x)--> ‡; x-->- ‡, f(x)-->- ‡