Video: Identify the Scatterplot That Is Modelled by the Given Equation

Which of the following scatter plots shows π‘Ž relationship that is appropriately modelled by the equation 𝑦 = π‘Žπ‘₯^𝑏, where π‘Ž is positive and 𝑏 is negative.

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Video Transcript

Which of the following scatter plots shows π‘Ž relationship that is appropriately modelled by the equation 𝑦 equals π‘Ž multiplied by π‘₯ to the power of 𝑏, where π‘Ž 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 π‘₯ increases, the value of 𝑦 decreases. Option B has no correlation. This means that there is no length between the variable π‘₯ 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 π‘₯ and 𝑦. As the value of π‘₯ 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 π‘šπ‘₯ plus 𝑏, where π‘š is the slope or gradient and 𝑏 is the 𝑦-intercept. This is not of the form 𝑦 equals π‘Ž multiplied by π‘₯ 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 π‘Ž multiplied by π‘₯ to the power of 𝑏. Option A represents exponential decay. As already mentioned, as the value of π‘₯ increases, the value of 𝑦 decreases.

Option C, on the other hand, represents exponential growth. As the value of π‘₯ 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 π‘Ž multiplied by π‘₯ to the 𝑏, where π‘Ž is positive and 𝑏 is negative.

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