Describe the end behavior of f(x)=2x^4-5x+1.
is "end behavior" x ---> ±∞ ?
then +∞ as x gets large in the positive direction
and -∞ as gets gets large in the negative direction
To describe the end behavior of a polynomial function, we consider the highest degree term in the function, which in this case is 2x^4. We can determine the end behavior by looking at the behavior of the leading term as x approaches positive infinity and negative infinity.
As x approaches negative infinity, 2x^4 becomes a large negative number, since negative values raised to an even power become positive. So, the function f(x) approaches negative infinity as x approaches negative infinity.
As x approaches positive infinity, 2x^4 becomes a large positive number, since positive values raised to an even power also become positive. So, the function f(x) approaches positive infinity as x approaches positive infinity.
Therefore, the end behavior of the function f(x) = 2x^4 - 5x + 1 can be described as follows:
As x approaches negative infinity, f(x) approaches negative infinity.
As x approaches positive infinity, f(x) approaches positive infinity.
To determine the end behavior of a polynomial function, such as the one given, f(x) = 2x^4 - 5x + 1, we need to examine the leading term, which is the term with the highest power of x.
In this case, the leading term is 2x^4. The degree of the polynomial is determined by the exponent of the leading term, which is 4.
Now, consider the sign of the coefficient of the leading term, which is positive (+2). This means that as x approaches positive infinity (+∞), the values of the function will also approach positive infinity. Additionally, as x approaches negative infinity (-∞), the values of the function will approach negative infinity.
In summary, the end behavior of the function f(x) = 2x^4 - 5x + 1 is as follows:
- As x approaches positive infinity (+∞), f(x) approaches positive infinity (+∞).
- As x approaches negative infinity (-∞), f(x) approaches negative infinity (-∞).