Give the end behavior of f(x) = −2x^5+4x^3+1 (1 point)
1. Up and Up
2. Down and Up
3. Down and Down
4. Up and Down
The end behavior of a polynomial function is determined by the degree and leading coefficient of the polynomial.
In this case, the degree of the polynomial function is 5, which is an odd number. Additionally, the leading coefficient is -2.
For odd-degree polynomial functions with a negative leading coefficient, the end behavior is "Down and Down", meaning that as x approaches positive infinity, the function approaches negative infinity, and as x approaches negative infinity, the function also approaches negative infinity.
Therefore, the correct answer is 3. Down and Down.
To determine the end behavior of the function f(x) = -2x^5 + 4x^3 + 1, we need to examine the powers and coefficients of the highest degree term.
In this case, the highest degree term is -2x^5. The coefficient of this term is -2, which is negative. Since the power of x is odd (5), the behavior will be opposite on each side of the graph.
Thus, the end behavior can be described as follows:
1. As x approaches positive infinity, f(x) will approach negative infinity. This is because the negative coefficient (-2) causes the graph to go down on the right side.
2. As x approaches negative infinity, f(x) will approach negative infinity. Since the power of x is odd, the graph will go up on the left side.
Therefore, the answer is option 3: Down and Down.
To determine the end behavior of a function, we look at the leading coefficient and the degree of the polynomial.
For the given function f(x) = -2x^5 + 4x^3 + 1, the leading coefficient is -2 and the degree is 5.
If the leading coefficient is positive and the degree is odd (as it is in this case), then the end behavior of the function is:
- As x approaches negative infinity, f(x) approaches negative infinity.
- As x approaches positive infinity, f(x) approaches positive infinity.
Therefore, the correct answer is:
1. Up and Up