Final answer:
The behavior of the function q(x) near its vertical asymptotes at x = 7 and x = -3 involves the function heading towards positive or negative infinity as x approaches these values. The horizontal asymptote of the function is y = 1, indicating the behavior of q(x) as x approaches positive or negative infinity. Sketching the function requires plotting these asymptotes and key points to visualize the trend of the function.
Step-by-step explanation:
The behavior of the function q(x) = (x+5)/((x-7)(x+3)) near its asymptotes can be determined by analyzing what happens to the function values as x approaches the asymptote values. The vertical asymptotes of this function are at x = 7 and x = -3 because the function is undefined when the denominator is zero (x-7=0 or x+3=0).
As x approaches 7 from the left (x < 7), q(x) goes to negative infinity. As x approaches 7 from the right (x > 7), q(x) goes to positive infinity. Similarly, as x approaches -3 from the left (x < -3), q(x) goes to positive infinity, and as it approaches from the right (x > -3), it goes to negative infinity.
The horizontal asymptote is analyzed by looking at the behavior of q(x) as x goes to positive or negative infinity. Since the degrees of the numerator and the denominator are the same, the horizontal asymptote is the ratio of the leading coefficients, which is 1. Thus, as x approaches infinity or negative infinity, the function approaches y = 1.
Sketch the graph of this function by plotting a few key points and the asymptotes, and then draw the function's curve in accordance with the behavior near the asymptotes.