Final answer:
The function h(x) has a leading term of -5x⁴, which determines its end behavior to decrease towards negative infinity as x approaches both positive and negative infinity. For the function f(x) at x = 3, y = x² fits the description of having a positive value with a positive, decreasing slope. Even functions are symmetric about the y-axis while odd functions have rotational symmetry around the origin.
Step-by-step explanation:
To describe the end behavior of the function h(x)=-5x⁴ + 7x³ - 6x² + 9x + 2, we look at the leading term. Since the function is a polynomial, its long-term behavior is dominated by the term with the highest degree, which in this case is -5x⁴. The negative coefficient indicates that as x goes to positive or negative infinity, h(x) will go to negative infinity. This is because the degree of the term, four, is even, which means the function will behave similarly at both ends.
For x = 3, a function f(x) with a positive value and a positive slope that is decreasing in magnitude indicates that the slope is positive but getting flatter. This rules out y = 13x because its slope is constant, not decreasing. On the other hand, the function y = x² has a positive slope that decreases in magnitude as x increases, which corresponds to f(x).
Even and odd functions have symmetries about the y-axis and the origin, respectively. An even function is symmetric about the y-axis, while an odd function has rotational symmetry about the origin, which can be visualized by reflecting the function first across the y-axis and then across the x-axis.