Which of the following scatter plots shows 𝑎 relationship that is appropriately modelled by the equation 𝑦 equals 𝑎 multiplied by 𝑥 to the power of 𝑏, where 𝑎 is positive and 𝑏 is negative.
Options A and D show a negative correlation, even though option D is a relatively weak negative correlation. In both of these cases, as the value of 𝑥 increases, the value of 𝑦 decreases. Option B has no correlation. This means that there is no length between the variable 𝑥 and the variable 𝑦. We can therefore say that there is no equation that models the relationship in scatter plot B. Option C demonstrates a positive correlation between 𝑥 and 𝑦. As the value of 𝑥 increases, so does the value of 𝑦. If we were to draw a line of best fit on option D, we would have a straight line. This means that option D shows a linear relationship.
The equation that models this would be of the type 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope or gradient and 𝑏 is the 𝑦-intercept. This is not of the form 𝑦 equals 𝑎 multiplied by 𝑥 to the power of 𝑏. We have, therefore, ruled out options B and D. Both options A and C demonstrate an exponential relationship. The lines of best fit are shown on the scatter plots. And the equations would be of the form 𝑦 equals 𝑎 multiplied by 𝑥 to the power of 𝑏. Option A represents exponential decay. As already mentioned, as the value of 𝑥 increases, the value of 𝑦 decreases.
Option C, on the other hand, represents exponential growth. As the value of 𝑥 increases, 𝑦 also increases. When dealing with exponential decay, our value of 𝑏, the power or exponent, is negative. Whereas with exponential growth, it is positive. We’re told in this question that 𝑏 is negative. This rules out option C. We can, therefore, conclude that the correct answer is option A. This is the scatter plot that shows a relationship that is modelled by the equation 𝑦 equals 𝑎 multiplied by 𝑥 to the 𝑏, where 𝑎 is positive and 𝑏 is negative.