Final Answer:
The correct answer to the question is b) An exponential curve.
Step-by-step explanation:
Let's go through each option to understand why:
a) A linear curve represents a constant rate of change. This means that the value increases or decreases by the same amount over equal intervals. A linear curve does not show a slowing expansion; rather, it shows a consistent expansion without any change in the rate of growth.
b) An exponential curve typically starts with a relatively small value and then increases at a rate proportional to its current value. As the value gets larger, the absolute increase also gets larger. However, an exponential curve that models a slowing expansion forever, such as in logistic growth, will start to increase at a slower rate as it approaches a limiting factor, never quite reaching a maximum value or carrying capacity. This is because as the curve approaches the limiting factor, the growth rate decays and the expansion slows down, but it never completely stops, thus seeming to expand forever at a decreasing rate.
c) A curve with a negative slope represents a decrease in value over time. As you move along the x-axis from left to right, the value of the function decreases, which is the opposite of what is described in the question. A curve with a negative slope cannot show an expansion.
d) A curve with a positive slope indicates an increase in value over time. However, this does not specify the type of increase (constant, increasing, or decreasing rate of change). Without additional information, a positive slope alone does not guarantee a slowing expansion forever.
Therefore, among the given options, the exponential curve (b) is the one that can represent a situation where the expansion slows without ever completely stopping, capturing the idea of continuous growth at a decreasing rate.