Final answer:
The end behavior of the polynomial function q(x) is such that as x approaches positive infinity, q(x) approaches positive infinity; and as x approaches negative infinity, q(x) approaches negative infinity.
Step-by-step explanation:
To determine the end behavior of the graph of the polynomial function q(x) = 3x7 + 5x5 - 8x4 - 12x2, we look at the leading term, which is 3x7. Since the degree (7) is odd and the leading coefficient (3) is positive, as x approaches positive infinity, q(x) will also approach positive infinity. Conversely, as x approaches negative infinity, q(x) will approach negative infinity. These behaviors describe the 'ends' of the graph on the x-axis.