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Problem-Solving and Data Analysis / Two-variable data: Models and scatterplots Difficulty: Hard

The scatterplot below shows the amount of electric energy generated, in millions of megawatt-hours, by nuclear sources over a 10‑year period.

The figure presents a scatterplot in the xy-plane titled “Electric Energy Generated by Nuclear Sources.” The x-axis is labeled “Time,” in years, and the numbers zero through 12, in increments of 2, are indicated. The y-axis is labeled “Energy,” in million megawatt-hours, and the numbers 700 through 820, in increments of 20, are indicated. There are 10 data points scattered in the form of a curve. The first data point begins approximately at the point with coordinates 1 comma 764. The points follow a path varying upward and downward and to the right, until reaching the point with coordinates of approximately 8 comma 806. The points then move downward and to the right, ending at the point with coordinates of approximately 10 comma 768. The data represented by the points are as follows. Note that all values are approximate. Point 1: 1 year, 764 million mega-watt hours. Point 2: 2 years, 788 million mega-watt hours. Point 3: 3 years, 781 million mega-watt hours. Point 4: 4 years, 785 million mega-watt hours. Point 5: 5 years, 804 million mega-watt hours. Point 6: 6 years, 803 million mega-watt hours. Point 7: 7 years, 796 million mega-watt hours. Point 8: 8 years, 806 million mega-watt hours. Point 9: 9 years, 790 million mega-watt hours. Point 10: 10 years, 768 million mega-watt hours.

 

Of the following equations, which best models the data in the scatterplot?

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Explanation

Choice D is correct. The data in the scatterplot roughly fall in the shape of a downward-opening parabola; therefore, the coefficient for the x squared term must be negative. Based on the location of the data points, the y-intercept of the parabola should be somewhere between 740 and 760. Therefore, of the equations given, the best model is y equals, negative 1 point 6 7 4, x squared, plus 19 point 7 6 x, plus 745 point 7 3.

Choices A and C are incorrect. The positive coefficient of the x squared term means that these equations each define upward-opening parabolas, whereas a parabola that fits the data in the scatterplot must open downward. Choice B is incorrect because it defines a parabola with a y-intercept that has a negative y-coordinate, whereas a parabola that fits the data in the scatterplot must have a y-intercept with a positive y-coordinate.