Final answer:
The appearance of a positive slope is an upward line from left to right, a negative slope is a downward line, and a zero slope is a horizontal line. For g(x) = 5x, there is only a positive slope and no interval with a negative or zero slope. For the function f(x), option b: y = x² could be correct as the slope is positive at x = 3 and decreases in magnitude as x increases.
Step-by-step explanation:
Positive slope, negative slope, and zero slope all refer to the steepness and direction of a line on a graph. A positive slope means the line is going upward from left to right, indicating that as the x-value increases, the y-value also increases.
On the other hand, a negative slope means the line is going downward from left to right, showing that as the x-value increases, the y-value decreases.
Lastly, a zero slope indicates a horizontal line, where the y-value remains constant regardless of the x-value.
Considering the function g(x) = 5x, its graph will be a straight line that rises from left to right, reflecting a positive slope throughout its domain. There is no interval where the function has a negative slope or a zero slope since the coefficient of x is positive (5), and it does not change.
The function f(x) at x = 3 with a positive value and a positive slope that is decreasing in magnitude with increasing x could be option b: y = x². This is because a quadratic function like x² has a positive slope that gets smaller (decreases in magnitude) as x increases past the vertex of the parabola (which in this case is at x=0).