Question

which of the following scatterplots shows a relationship that is appropriately modeled with the equation y = ax^b, where a is positive and b is negative? a) scatter - plot with points showing a decreasing trend b) scatter - plot with points showing a decreasing trend c) scatter - plot with points showing an increasing trend d) scatter - plot with points showing no clear trend

Explanation:

Step1: Analyze the function properties

For the function $y = ax^{b}$ with $a>0$ and $b < 0$, as $x$ increases, $y$ decreases. Also, the rate of decrease slows down.

Step2: Examine each scatter - plot

• Option A: The points show a decreasing trend as $x$ increases, and the rate of decrease slows down, which is consistent with $y = ax^{b}$ where $a>0$ and $b < 0$.
• Option B: The points seem to decrease rapidly at first and then level off in a way that is not consistent with the power - law decay of $y = ax^{b}$ with $b<0$.
• Option C: The points show an increasing trend as $x$ increases, which is not consistent with $y = ax^{b}$ where $b < 0$.
• Option D: The points are randomly scattered and do not show a clear decreasing trend as required for $y = ax^{b}$ with $b < 0$.

Answer:

A