The figure shown represents a relation between the concentration of a drug and the duration of time after taking it. Which of the following regression types would be the best model for this data? A) A linear function. B) A polynomial function. C) A quadratic function. Or D) An exponential function.
The graph shows us that as the time increases, the concentration decreases. This in itself is not enough to answer the question. We need to consider what all four of the functions would look like.
A linear function is a straight-line function of the form 𝑦 equals 𝑚𝑥 plus 𝑏. It can slope upwards or downwards. As the line of best fit on our graph would not be a straight line, then this is not the best model for the data.
A quadratic function would be a U-shape or n-shape parabola. Our equation for a quadratic function is 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐. These parabolas would either have one maximum or minimum point as shown in pink. As the concentration continues to decrease over time in our case, a quadratic function would not be the best model either.
Linear functions and quadratic functions are types of polynomial functions. The word polynomial means consisting of several terms. As a polynomial function could be a linear or quadratic function, this is not the best model for the data.
As we have ruled out options A, B, and C, this suggests that the correct answer is an exponential function. Exponential functions are curves that tend towards a specific value, usually infinity or zero. Exponential growth increases over time, whereas exponential decay decreases over time. We can see from our graph that the concentration decreases over time. It is an example of exponential decay.
Our line of best fit decreases rapidly to start with and then levels out over time, tending towards zero. This means that the relation between the concentration of the drug and the duration of time after taking it would be best modeled by an exponential function.