Question

Insect Population (a) Suppose an insect population increases by a constant number each month. Explain why the number of insects can be represented by a linear function. (b) Suppose an insect population increases by a constant percentage each month. Explain why the number of insects can be represented by an exponential function.

Short answer

For part (a), the insect population can be represented by a linear function because the number of insects increases by a consistent amount each month, mirroring the constant rate of change in linear functions. For part (b), the insect population can be represented by an exponential function, as the growth rate decided by a fixed percentage is dependent on the current population, mimicking the growth behaviour of exponential functions.

Step-by-step solution

Explanation of Linear Function

A linear function or a straight line function has a constant rate of change. The rate of change is the same across the function. In other words, the change between y-values is proportional to the change between x-values. A linear function can be represented by the formula y = mx+b, where m is the rate of change and b is the y-intercept. Hence, if the insect population increases by a constant number each month, it is akin to a linear function where the step up on y-axis per month (the rate of change) is constant.

Explanation of Exponential Function

An exponential function is a function in which the variable is in the exponent and it can be represented in the general form f(x) = a.b^x + c, where a, b and c are constants, and b>0, b ≠1. It has the property that the rate of growth is proportional to the function's current value, which means as the insect population increases, the rate at which it grows also increases. Therefore, if the insect population increases by a constant percentage each month, it is akin to an exponential function since the growth rate is directly proportional to the size of the population.