Final answer:
To determine the end behavior of the function f(x) = 4 + (1/4)x^4 - (1/3)x^3, the leading term with the highest power, (1/4)x^4, indicates that as x approaches both positive and negative infinity, the function will approach positive infinity, resulting in a U-shaped graph.
Step-by-step explanation:
The question is asking to determine the end behavior of the polynomial function f(x) = 4 + (1/4)x4 - (1/3)x3. The end behavior of a polynomial is dictated by its leading term, which is the term with the highest power of x. In this function, the leading term is (1/4)x4, which is a positive coefficient with an even exponent. Therefore, as x approaches infinity, f(x) will also approach infinity, and as x approaches negative infinity, f(x) will again approach infinity, since any even power of a negative number is positive.To graph such a polynomial, one would expect the ends of the graph to rise in both the left and right direction, making a U-shape. This behavior indicates that the correct end behavior diagram would show the graph heading upwards in both directions. You can visualize this as a smooth curve similar in shape to a parabola that opens upwards, except that a fourth-degree polynomial may have up to three turning points whereas a parabola has just one.
To determine the end behavior of the function f(x), we need to consider the highest power of x in the equation. In this case, the highest power of x is 4. Since the coefficient of x^4 is positive (+1/4), the end behavior of the function is as follows:As x approaches positive infinity, f(x) approaches positive infinity, meaning the graph rises to the right. As x approaches negative infinity, f(x) also approaches positive infinity, indicating that the graph rises to the left.Therefore, the correct end behavior diagram for f(x)=4+(1)/(4)x^(4)-(1)/(3)x^(3) is Option B with the graph rising on both sides.