Final answer:
The equation f(x) = 3x^12 +6x will result in a graph with both ends trending upwards, due to the positive coefficient and even exponent of the highest degree term.
Step-by-step explanation:
The equation given is f(x) = 3x12 +6x. To understand the behavior of the graph at the ends, we look at the highest degree term, which in this case is x12. Since the coefficient of this term, 3, is positive and the exponent 12 is an even number, both ends of the graph will go upwards. No matter the value of x, whether it is positive or negative, the value of x12 will always be positive (since any number to an even power is positive), and when multiplied by 3, it will remain positive. As x becomes very large or very small (x approaches positive or negative infinity), the 3x12 term will dominate the 6x term, and the graph will resemble the graph of y=3x12, which rises steeply on both ends. Thus, at both ends of the graph, as x approaches infinity or negative infinity, the value of f(x) will approach infinity, leading to an upward trend on both sides.