Two variable Data - Models and Scatter Plots

Choice C is correct. According to the line of best fit, a planetoid with a distance from the Sun of 1.2 AU has a predicted density between and . The only choice in this range is 4.6.Choices A, B, and D are incorrect and may result from misreading the information in the scatterplot.

Choice C is correct. For each point on the scatterplot shown, the x-value represents the weight, in pounds, of a female gray wolf and the y-value represents the number of offspring that wolf produced. The point on the graph with an x-value of \(50\) has a y-value of \(6\). Therefore, the \(50\)-pound gray wolf produced \(6\) offspring. Choice A is incorrect. One of the wolves produced \(8\) offspring, but its weight was greater than \(50\) pounds. Choice B is incorrect. Three of the wolves produced \(7\) offspring each, but their weights were each greater than \(50\) pounds. Choice D is incorrect. Two of the wolves produced \(5\) offspring each, but their weights were each less than \(50\) pounds.

Choice C is correct. Because the value of the investment increases each year, the function that best models how the value of the investment changes over time is an increasing function. It′s given that each year, the value of the investment increases by \(0.49 percent sign\) of its value the previous year. Since the value of the investment changes by a fixed percentage each year, the function that best models how the value of the investment changes over time is an exponential function. Therefore, the function that best models how the value of the investment changes over time is an increasing exponential function. Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice C is correct. It's given that the scatterplot shows the relationship between two variables, \(x\) and \(y\), and a line of best fit is shown. For the line of best fit shown, for each increase in the value of \(x\) by \(1\), the corresponding value of \(y\) increases by a constant rate. It follows that the relationship between the variables \(x\) and \(y\) has a positive linear trend. A line in the xy-plane that passes through the points \((a, b)\) and \((c, d)\) has a slope of \(d -b/c -a\). The line of best fit shown passes approximately through the points \((0, 0.25)\) and \((4, 2)\). It follows that the slope of this line is approximately \(2 -0.25/4 -0\), which is equivalent to \(0.4375\). Therefore, of the given choices, \(0.44\) is closest to the slope of the line of best fit shown. Choice A is incorrect. This is the slope of a line of best fit for a relationship between \(x\) and \(y\) that has a -, rather than a positive, linear trend. Choice B is incorrect. This is the slope of a line of best fit for a relationship between \(x\) and \(y\) that has a -, rather than a positive, linear trend. Choice D is incorrect and may result from conceptual or calculation errors.

Choice C is correct. The predicted increase in total fat, in grams, for every increase of 1 gram in total protein is represented by the slope of the line of best fit. Any two points on the line can be used to calculate the slope of the line as the change in total fat over the change in total protein. For instance, it can be estimated that the points and are on the line of best fit, and the slope of the line that passes through them is, or 1.4. Of the choices given, 1.5 is the closest to the slope of the line of best fit.Choices A, B, and D are incorrect and may be the result of incorrectly finding ordered pairs that lie on the line of best fit or of incorrectly calculating the slope.

Choice B is correct. A line in the xy-plane that passes through points \((x_1, y_1)\) and \((x_2, y_2)\) has a slope of \(y_2 -y_1/x_2 -x_1\). The line of best fit shown passes approximately through the points \((0, 14)\) and \((13, 0)\). It follows that the slope of this line of best fit is approximately \( -14/13 -0\), or \(-14/13\). Of the given choices, \(-1.1\) is closest to \(-14/13\). Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The data points suggest that as the variable \(x\) increases, the variable \(y\) decreases, which implies that an appropriate linear model for the data has a -slope. The data points also show that when \(x\) is close to \(0\), \(y\) is greater than \(9\). Therefore, the y-intercept of the graph of an appropriate linear model has a y-coordinate greater than \(9\). The graph of an equation of the form \(y = a + b x\), where \(a\) and \(b\) are constants, has a y-intercept with a y-coordinate of \(a\) and has a slope of \(b\). Of the given choices, only choice D represents a graph that has a -slope, \(-0.9\), and a y-intercept with a y-coordinate greater than \(9\), \(9.4\). Choice A is incorrect. The graph of this equation has a positive slope, not a -slope, and a y-intercept with a y-coordinate less than \(1\), not greater than \(9\). Choice B is incorrect. The graph of this equation has a y-intercept with a y-coordinate less than \(1\), not greater than \(9\). Choice C is incorrect. The graph of this equation has a positive slope, not a -slope.

Choice B is correct. In the given scatterplot, the x-values represent the distance above sea level, in feet, and the y-values represent the temperature, in \(° F\). The point on the line of best fit with an x-value of \(4,000\) has a corresponding y-value of \(35\). Therefore, at a distance of \(4,000\) feet above sea level, the temperature predicted by the line of best fit is \(35 ° F\). Choice A is incorrect. This is the temperature, in \(° F\), predicted by the line of best fit at a distance of \(0\) feet above sea level. Choice C is incorrect. This is the measured temperature, in \(° F\), at a distance of \(6,000\) feet above sea level. Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. Account A starts with $500 and earns interest at 6% per year, so in the first year Account A earns (500)(0.06) = $30, which is greater than the $25 that Account B earns that year. Compounding interest can be modeled by an increasing exponential function, so each year Account A will earn more money than it did the previous year. Therefore, each year Account A earns at least $30 in interest. Since Account B always earns $25 each year, Account A always earns more money per year than Account B.Choices B and D are incorrect. Account A earns $30 in the first year, which is greater than the $25 Account B earns in the first year. Therefore, neither the statement that Account A always earns less money per year than Account B nor the statement that Account A earns less money than Account B at first can be true. Choice C is incorrect. Since compounding interest can be modeled by an increasing exponential function, each year Account A will earn more money than it did the previous year. Therefore, Account A always earns at least $30 per year, which is more than the $25 per year that Account B earns.

Choice D is correct. It's given that for \(x > 0\), \(f(x)\) is equal to \(201 percent sign\) of \(x\). This is equivalent to \(f(x)= 201/100 x\), or \(f(x)= 2.01 x\), for \(x > 0\). This function indicates that as \(x\) increases, \(f(x)\) also increases, which means \(f\) is an increasing function. Furthermore, \(f(x)\) increases at a constant rate of \(2.01\) for each increase of \(x\) by \(1\). A function with a constant rate of change is linear. Thus, the function \(f\) can be described as an increasing linear function. Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors. Choice C is incorrect. This could describe the function \(f(x)=(2.01)Superscript x\), where \(f(x)\) is equal to \(201 percent sign\) of \(f(x -1)\), not \(x\), for \(x > 0\).

Choice C is correct. At \(x = 32\), the line of best fit has a y-value between \(2\) and \(3\). The only choice with a value between \(2\) and \(3\) is choice C. Choice A is incorrect. This is the difference between the y-value predicted by the line of best fit and the actual y-value at \(x = 32\) rather than the y-value predicted by the line of best fit at \(x = 32\). Choice B is incorrect. This is the y-value predicted by the line of best fit at \(x = 31\) rather than at \(x = 32\). Choice D is incorrect. This is the y-value predicted by the line of best fit at \(x = 33\) rather than at \(x = 32\).

Choice B is correct. From the shape of the cluster of points, the line of best fit should pass roughly through the points and . Therefore, these two points can be used to find an approximate equation for the line of best fit. The slope of this line of best fit is therefore, or 100. The equation for the line of best fit, in slope-intercept form, is for some value of b. Using the point, 1 can be substituted for x and 200 can be substituted for y:, or . Substituting this value into the slope-intercept form of the equation gives .Choice A is incorrect. The line defined by passes through the points and, both of which are well above the cluster of points, so it cannot be a line of best fit. Choice C is incorrect. The line defined by passes through the points and, both of which lie at the bottom of the cluster of points, so it cannot be a line of best fit. Choice D is incorrect and may result from correctly calculating the slope of a line of best fit but incorrectly assuming the y-intercept is at .

Choice B is correct. The given values show that as \(x\) increases, \(f(x)\) also increases, which means that \(f\) is an increasing function. Furthermore, \(f(x)\) increases at a constant rate of \(1\) for each increase of \(x\) by \(1\). A function with a constant rate of change is linear. Thus, the function \(f\) can be described as an increasing linear function. Choice A is incorrect. For a decreasing linear function, as \(x\) increases, \(f(x)\) decreases rather than increases. Choice C is incorrect. For a decreasing exponential function, for each increase of \(x\) by \(1\), \(f(x)\) decreases by a fixed percentage rather than increases at a constant rate. Choice D is incorrect. For an increasing exponential function, for each increase of \(x\) by \(1\), \(f(x)\) increases by a fixed percentage rather than at a constant rate.

Choice B is correct. The association between the elevation and the temperature of locations in the Lake Tahoe Basin can be described by looking at the direction of the line of best fit. The line of best fit slopes downward, which corresponds to the temperature decreasing as the elevation increases.Choices A and C are incorrect. Both of these choices would be represented by a line of best fit that slopes from the lower left to the right of the graph, which isn’t what’s shown on the graph. Choice D is incorrect. This choice would be represented by a line of best fit that is horizontal or has a slope very close to 0. This is not what’s shown on the graph.

Choice C is correct. A line in the xy-plane that passes through points \((x_1, y_1)\) and \((x_2, y_2)\) has a slope of \(y_2 -y_1/x_2 -x_1\). The line of best fit shown passes approximately through the points \((0, 0.2)\) and \((5, 9.3)\). It follows that the slope of this line is approximately \(9.3 -0.2/5 -0\), which is equivalent to \(9.1/5\), or \(1.82\). Therefore, of the given choices, \(1.8\) is closest to the slope of the line of best fit shown. Choice A is incorrect. This value is closest to the y-coordinate of the y-intercept of the line of best fit shown. Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice B is correct. The equation representing a linear model can be written in the form \(y = a + b x\), or \(y = b x + a\), where \(b\) is the slope of the graph of the model and \((0, a)\) is the y-intercept of the graph of the model. The scatterplot shows that as the x-values of the data points increase, the y-values of the data points decrease, which means the graph of an appropriate linear model has a -slope. Therefore, \(b < 0[/latex]. The scatterplot also shows that the data points are close to the y-axis at a positive value of [latex]y[/latex]. Therefore, the y-intercept of the graph of an appropriate linear model has a positive y-coordinate, which means [latex]a > 0\). Of the given choices, only choice B, \(y = -1.9 x + 10.1\), has a -value for \(b\), the slope, and a positive value for \(a\), the y-coordinate of the y-intercept. Choice A is incorrect. The graph of this model has a y-intercept with a -y-coordinate, not a positive y-coordinate. Choice C is incorrect. The graph of this model has a positive slope, not a -slope, and a y-intercept with a -y-coordinate, not a positive y-coordinate. Choice D is incorrect. The graph of this model has a positive slope, not a -slope.

Correct Answer Rationale Choice C is correct. According to the model, the stopπng distance, in feet, of a vehicle traveling 55 miles per hour is about 200 feet. Of the choices given, the best estimate of the stopπng distance for a car traveling 55 miles per hour is 210 feet.Incorrect Answer Rationale Choices A, B, and D are incorrect and may be the result of incorrectly reading the given quadratic model. The corresponding x-values to the y-values of 25 and 30 are not part of the model. The corresponding x-value to a y-value of 250 is approximately 60 mph, not 55 mph.

Choice C is correct. The scatterplot shows that there was an increase in recorded temperature from \(x = 2\) to \(x = 3\) and from \(x = 6\) to \(x = 7\). When \(x = 2\), the recorded temperature was approximately \(60 ° F\) and when \(x = 3\), the recorded temperature was greater than \(70 ° F\). This means that the increase in recorded temperature from \(x = 2\) to \(x = 3\) was greater than \((70 -60)° F\), or \(10 ° F\). When \(x = 6\), the recorded temperature was greater than \(60 ° F\) and when \(x = 7\), the recorded temperature was less than \(70 ° F\). This means that the increase in recorded temperature from \(x = 6\) to \(x = 7\) was less than \((70 -60)° F\), or \(10 ° F\). It follows that the greatest increase in recorded temperature took place from \(x = 2\) to \(x = 3\). Choice A is incorrect. The increase in recorded temperature from \(x = 6\) to \(x = 7\) was less than the increase in recorded temperature from \(x = 2\) to \(x = 3\). Choice B is incorrect. From \(x = 5\) to \(x = 6\), a decrease, not an increase, in recorded temperature took place. Choice D is incorrect. From \(x = 1\) to \(x = 2\), a decrease, not an increase, in recorded temperature took place.

Choice A is correct. The data show a strong linear relationship between x and y. The line of best fit for a set of data is a linear equation that minimizes the distances from the data points to the line. An equation for the line of best fit can be written in slope-intercept form,, where m is the slope of the graph of the line and b is the y-coordinate of the y-intercept of the graph. Since, for the data shown, the y-values increase as the x-values increase, the slope of a line of best fit must be positive. The data shown lie almost in a line, so the slope can be roughly estimated using the formula for slope, . The leftmost and rightmost data points have coordinates of about and, so the slope is approximately, which is a little greater than 3. Extension of the line to the left would intersect the y-axis at about . Only choice A represents a line with a slope close to 3 and a y-intercept close to .Choice B is incorrect and may result from switching the slope and y-intercept. The line with a y-intercept of and a slope of 0.8 is farther from the data points than the line with a slope of 3 and a y-intercept of . Choices C and D are incorrect. They represent lines with -slopes, not positive slopes.

Choice D is correct. The data in the scatterplot roughly fall in the shape of a downward-opening parabola; therefore, the coefficient for the term must be -. 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 .Choices A and C are incorrect. The positive coefficient of the 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 -y-coordinate, whereas a parabola that fits the data in the scatterplot must have a y-intercept with a positive y-coordinate.

The correct answer is 480. An average temperature of corresponds to the value 32 on the x-axis. On the line of best fit, an x-value of 32 corresponds to a y-value of 480. The values on the y-axis correspond to the number of people predicted to visit this beach. Therefore, 480 people are predicted to visit this beach on a day with an average temperature of .

Choice A is correct. The graph of function \(f\) shows that as \(x\) increases, \(f(x)\) also increases, which means \(f(x)\) is an increasing function. The graph of \(f\) is a line, which indicates a constant rate of change. A function that has a constant rate of change is a linear function. Therefore, function \(f\) can be described as increasing linear. Choice B is incorrect. For a decreasing function, as \(x\) increases, \(f(x)\) decreases, rather than increases. Choice C is incorrect. For a decreasing function, as \(x\) increases, \(f(x)\) decreases, rather than increases, and the graph of an exponential function isn't a line. Choice D is incorrect. The graph of an exponential function isn't a line.

Choice B is correct. For the given line graph, the estimated number of chipmunks is represented on the vertical axis. The greatest estimated number of chipmunks in the state park is indicated by the greatest height in the line graph. This height is achieved when the year is \(1994\). Choice A is incorrect and may result from conceptual errors. Choice C is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice D is correct. A line in the xy-plane that passes through the points \((x_1, y_1)\) and \((x_2, y_2)\) has a slope of \(y_2 -y_1/x_2 -x_1\). The line of best fit shown passes approximately through the points \((1, 3.3)\) and \((7, 14.5)\). It follows that the slope of this best fit line is approximately \(14.5 -3.3/7 -1\), which is equivalent to \(11.2/6\), or approximately \(1.87\). Therefore, of the given choices, \(2\) is closest to the slope of the line of best fit shown. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is correct. The relationship between the values of x and their corresponding y-values is nonlinear if the rate of change between these pairs of values isn’t constant. The table for choice D gives four pairs of values:,,, and . Finding the rate of change, or slope, between and by using the slope formula,, yields, or 6. Finding the rate of change between and yields, or 12. Finding the rate of change between and yields, or 24. Since the rate of change isn’t constant for these pairs of values, this table shows a nonlinear relationship.Choices A, B, and C are incorrect. The rate of change between the values of x and their corresponding y-values in each of these tables is constant, being 3, 4, and 5, respectively. Therefore, each of these tables shows a linear relationship.

The correct answer is \(40\). For each data point on the scatterplot, the x-value represents the growing season after transplantation and the y-value represents the weight, in pounds, of all nectarines that grew on the tree during the season. The scatterplot shows a data point at \((4, 40)\). It follows that during the \(4^{th}\) growing season after the tree’s transplantation, \(40\) pounds of nectarines grew on the tree.

Choice B is correct. According to the graph, when the car is traveling at 50 miles per hour, the braking distance is approximately 225 feet, and when the car is traveling at 30 miles per hour, the braking distance is approximately 100 feet. The difference between these braking distances is, or 125 feet.Choice A is incorrect and may result from finding the braking distance for 20 miles per hour, the difference between the given speeds. Choice C is incorrect and may result from subtracting the speed from the braking distance at 50 miles per hour. Choice D is incorrect and may result from finding the difference in the braking distances at 60 and 20 miles per hour.

Choice A is correct. A line of best fit for the data in a scatterplot is a line that follows the trend of the data with approximately half the data points above and half the data points below the line. Based on the given data, a line of best fit will have a positive y-intercept on or near the point and a -slope. All of the choices are in slope-intercept form, where m is the slope and b is the y-coordinate of the y-intercept. Only choice A is an equation of a line with a positive y-intercept at and a -slope, .Choice B is incorrect. This equation is for a line that has a -y-intercept, not a positive y-intercept. Choices C and D are incorrect and may result from one or more sign errors and from switching the values of the y-intercept and the slope in the equation.

The correct answer is 285. The number of people predicted to visit the beach each day is represented by the y-values of the line of best fit, and the average temperature, in °s Celsius (), is represented by the x-values. Since the slope of the line of best fit is approximately 57, the y-value, or the number of people predicted to visit the beach each day, increases by 57 for every x-value increase of 1, or every increase in average temperature. Therefore, an increase of in average temperature corresponds to a y-value increase of additional people per day predicted to visit the beach.

Choice B is correct. It′s given that the airplane descends at a constant rate of \(400 feet per minute\). Since the altitude decreases by a constant amount during each fixed time period, the relationship between the airplane′s altitude and time is linear. Since the airplane descends from an altitude of \(9,500 feet\) to \(5,000 feet\), the airplane′s altitude is decreasing with time. Thus, the relationship is best modeled by a decreasing linear function. Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. The line of best fit shown has a positive slope and intersects the y-axis at a positive y-value. The graph of an equation of the form \(y = m x + b\), where \(m\) and \(b\) are constants, has a slope of \(m\) and intersects the y-axis at a y-value of \(b\). Of the given choices, only \(y = x + 3.4\) represents a line that has a positive slope, \(1\), and intersects the y-axis at a positive y-value, \(3.4\). Choice B is incorrect. This equation represents a line that intersects the y-axis at a -y-value, not a positive y-value. Choice C is incorrect. This equation represents a line that has a -slope, not a positive slope. Choice D is incorrect. This equation represents a line that has a -slope, not a positive slope, and intersects the y-axis at a -y-value, not a positive y-value.

Choice C is correct. On the line of best fit, an x-value of \(1,200\) corresponds to a y-value between \(10\) and \(12\). Therefore, of the given choices, \(11\) is closest to the y-value predicted by the line of best fit at \(x = 1,200\). Choice A is incorrect. This is the integer value closest to the y-value predicted by the line of best fit at \(x = 1,800\). Choice B is incorrect. This is the integer value closest to the y-value predicted by the line of best fit at \(x = 1,500\). Choice D is incorrect. This is the integer value closest to the y-value predicted by the line of best fit at \(x = 600\).

Choice A is correct. An equation of a line of best fit can be written in the form, where a is the y-intercept of the line and b is the slope. In the scatterplot shown, the line of best fit intersects the y-axis just over halfway between 10,000 and 20,000, or approximately 16,000. The line of best fit also intersects the graph at . Using the slope formula and two points that lie on the graph such as and, the slope can be approximated as, or 4,800. Only choice A has a y-intercept near the estimate of 16,000 and a slope near the estimate of 4,800. Therefore, an equation of the line of best fit could be .Choice B is incorrect because the values for the slope and the y-coordinate of the y-intercept are switched. Choice C is incorrect because the value for the slope is approximately double the actual slope. Choice D is incorrect because the values for the slope and the y-intercept are switched and because the slope is approximately double the actual slope.

Choice D is correct. A linear model can be written in the form \(y = m x + b\), where \(m\) is the slope of the graph of the model in the xy-plane and \((0, b)\) is the y-intercept. The graph of an appropriate linear model for this relationship passes near the points \((1, 3)\) and \((9, 10)\) in the xy-plane. Two points on a line, \((x_1, y_1)\) and \((x_2, y_2)\), can be used to find the slope of the line using the slope formula, \(m = y_2 -y_1/x_2 -x_1\). Substituting the points \((1, 3)\) and \((9, 10)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in the slope formula yields \(m = 10 -3/9 -1\), or \(m = 0.875\). Therefore, the value of \(m\) for an appropriate linear model is approximately \(0.875\). Substituting \(0.875\) for \(m\) in \(y = m x + b\) yields \(y = 0.875 x + b\). Since an appropriate linear model passes near the point \((1, 3)\), the approximate value of \(b\) can be found by substituting \(1\) for \(x\) and \(3\) for \(y\) in the equation \(y = 0.875 x + b\), which yields \(3 =(0.875)(1)+ b\), or \(3 = 0.875 + b\). Subtracting \(0.875\) from both sides of this equation yields \(2.125 = b\). Therefore, the value of \(b\) for an appropriate linear model is approximately \(2.125\). Thus, of the given choices, \(y = 0.9 x + 2.2\) is the most appropriate linear model for this relationship.Alternate approach: A linear model can be written in the form \(y = m x + b\), where \(m\) is the slope of the graph of the model in the xy-plane and \((0, b)\) is the y-intercept. The scatterplot shows that as the x-values of the data points increase, the y-values of the data points increase, which means the graph of an appropriate linear model has a positive slope. Of the given choices, \(y = 0.9 x + 2.2\) is the only linear model whose graph has a positive slope. Choice A is incorrect. The graph of this model has a -slope, not a positive slope. Choice B is incorrect. The graph of this model has a -slope, not a positive slope. Choice C is incorrect. The graph of this model has a -slope, not a positive slope.

The correct answer is \(6\). The line of best fit predicts a greater y-value than the actual y-value for any data point that's located below the line of best fit. For the scatterplot shown, \(6\) of the data points are below the line of best fit. Therefore, the line of best fit predicts a greater y-value than the actual y-value for \(6\) of the data points.

The correct answer is \(9 halves\). It's given that the scatterplot shows the relationship between the length of time \(y\), in hours, a certain bird spent in flight and the number of days after January \(11\), \(x\). Since January \(13\) is \(2\) days after January \(11\), it follows that January \(13\) corresponds to an x-value of \(2\) in the scatterplot. In the scatterplot, when \(x = 2\), the corresponding value of \(y\) is \(6\). In other words, on January \(13\), the bird spent \(6\) hours in flight. Since January \(15\) is \(4\) days after January \(11\), it follows that January \(15\) corresponds to an x-value of \(4\) in the scatterplot. In the scatterplot, when \(x = 4\), the corresponding value of \(y\) is \(15\). In other words, on January \(15\), the bird spent \(15\) hours in flight. Therefore, the average rate of change, in hours per day, of the length of time the bird spent in flight on January \(13\) to the length of time the bird spent in flight on January \(15\) is the difference in the length of time, in hours, the bird spent in flight divided by the difference in the number of days after January \(11\), or \(15 -6/4 -2\), which is equivalent to \(9 halves\). Note that 9/2 and 4.5 are examples of ways to enter a correct answer.

Choice D is correct. The change in the number of 3-D movies released between any two consecutive years can be found by first estimating the number of 3-D movies released for each of the two years and then finding the positive difference between these two estimates. Between 2003 and 2004, this change is approximately movies; between 2008 and 2009, this change is approximately movies; between 2009 and 2010, this change is approximately movies; and between 2010 and 2011, this change is approximately movies. Therefore, of the pairs of consecutive years in the choices, the greatest increase in the number of 3-D movies released occurred during the time period between 2010 and 2011.Choices A, B, and C are incorrect. Between 2010 and 2011, approximately 20 more 3-D movies were released. The change in the number of 3-D movies released between any of the other pairs of consecutive years is significantly smaller than 20.

Choice C is correct. The data in the scatterplot show an increasing linear trend. The density when the juice concentration is 60% will be between the densities shown at about 53% and 67% concentration, or between about 1,255 and 1,340 kg/m3. Of the choices given, only 1,300 falls within this range.Choices A, B, and D are incorrect. These are the approximate densities of grape juice with a concentration of 45%, 55%, and 70%, respectively.

Choice C is correct. A line that most closely fits the data is a line with an approximately balanced number of data points above and below the line. Fitting a line to the data shown results in a line with an approximate slope of 3 and a y-intercept near the point . An equation for the line can be written in slope-intercept form,, where m is the slope and b is the y-coordinate of the y-intercept. The equation in choice C fits the data most closely.Choices A and B are incorrect because the slope of the lines of these equations is 0.82, which is a value that is too small to be the slope of the line that fits the data shown. Choice D is incorrect. The graph of this equation has a y-intercept at, not . This line would lie below all of the data points, and therefore would not closely fit the data.

Choice C is correct. For the line graph shown, the probability of snow, as a percent, is represented on the vertical axis. According to the line graph, during this four-day period, the probability of snow is \(30 percent sign\) for Thursday. Choice A is incorrect. The probability of snow on Tuesday is \(60 percent sign\). Choice B is incorrect. The probability of snow on Wednesday is \(90 percent sign\). Choice D is incorrect. The probability of snow on Friday is \(70 percent sign\).

Choice D is correct. On the line of best fit, d increases from approximately 480 to 880 between and . The slope of the line of best fit is the difference in d-values divided by the difference in t-values, which gives, or approximately 33. Writing the equation of the line of best fit in slope-intercept form gives, where b is the y-coordinate of the y-intercept. This equation is satisfied by all points on the line, so when . Thus,, which is equivalent to . Subtracting 396 from both sides of this equation gives . Therefore, an equation for the line of best fit could be .Choice A is incorrect and may result from an error in calculating the slope and misidentifying the y-coordinate of the y-intercept of the graph as the value of d at rather than the value of d at . Choice B is incorrect and may result from using the smallest value of t on the graph as the slope and misidentifying the y-coordinate of the y-intercept of the graph as the value of d at rather than the value of d at . Choice C is incorrect and may result from misidentifying the y-coordinate of the y-intercept as the smallest value of d on the graph.

Choice C is correct. Any data point that's located above the line of best fit has a y-value that's greater than the y-value predicted by the line of best fit. For the scatterplot shown, \(6\) of the data points are above the line of best fit. Therefore, \(6\) of the data points have an actual y-value that's greater than the y-value predicted by the line of best fit. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect. This is the number of data points that have an actual y-value that's less than the y-value predicted by the line of best fit. Choice D is incorrect and may result from conceptual or calculation errors.

Choice B is correct. The charity with the greatest percent of total expenses spent on programs is represented by the highest point on the scatterplot; this is the point that has a vertical coordinate slightly less than halfway between 90 and 95 and a horizontal coordinate slightly less than halfway between 3,000 and 4,000. Thus, the charity represented by this point has a total income of about $3,400 million and spends about 92% of its total expenses on programs. The percent predicted by the line of best fit is the vertical coordinate of the point on the line of best fit with horizontal coordinate $3,400 million; this vertical coordinate is very slightly more than 85. Thus, the line of best fit predicts that the charity with the greatest percent of total expenses spent on programs will spend slightly more than 85% on programs. Therefore, the difference between the actual percent (92%) and the prediction (slightly more than 85%) is slightly less than 7%.Choice A is incorrect. There is no charity represented in the scatterplot for which the difference between the actual percent of total expenses spent on programs and the percent predicted by the line of best fit is as much as 10%. Choices C and D are incorrect. These choices may result from misidentifying in the scatterplot the point that represents the charity with the greatest percent of total expenses spent on programs.

Choice A is correct. The line of best fit shown intersects the y-axis at a positive y-value and has a positive slope. The graph of an equation of the form \(y = a + b x\), where \(a\) and \(b\) are constants, intersects the y-axis at a y-value of \(a\) and has a slope of \(b\). Of the given choices, only choice A represents a line that intersects the y-axis at a positive y-value, \(2.8\), and has a positive slope, \(1.7\). Choice B is incorrect. This equation represents a line that has a -slope, not a positive slope. Choice C is incorrect. This equation represents a line that intersects the y-axis at a -y-value, not a positive y-value. Choice D is incorrect. This equation represents a line that intersects the y-axis at a -y-value, not a positive y-value, and has a -slope, not a positive slope.

Choice B is correct. Since the work is done at a constant rate, a linear model best models the situation. The number of shirts remaining is dependent on the length of time the inspector has worked; therefore, if the relationship were graphed, time would be the variable of the horizontal axis and the number of remaining shirts would be the variable of the vertical axis. Since the number of shirts decreases as the time worked increases, it follows that the slope of this graph is -.Choice A is incorrect and may result from incorrectly reasoning about the slope. Choices C and D are incorrect and may result from not identifying the constant rate of work as a characteristic of a linear model.

Choice C is correct. On the line of best fit, an x-value of \(25.5\) corresponds to a y-value between \(8\) and \(8.5\). Therefore, at \(x = 25.5\), \(8.2\) is closest to the y-value predicted by the line of best fit. Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice B is correct. Theresa’s speed was increasing from 0 to 5 minutes and from 20 to 25 minutes, which is a total of 10 minutes. Theresa’s speed was decreasing from 10 minutes to 20 minutes and from 25 to 30 minutes, which is a total of 15 minutes. Therefore, Theresa’s speed was NOT increasing for a longer period of time than it was decreasing.Choice A is incorrect. Theresa ran at a constant speed for the 5-minute period from 5 to 10 minutes. Choice C is incorrect. Theresa’s speed decreased at a constant rate during the last 5 minutes, which can be seen since the graph is linear during that time. Choice D is incorrect. Theresa’s speed reached its maximum at 25 minutes, which is within the last 10 minutes.

Choice A is correct. The year with the fewest graduates who enrolled in college within 24 months of graduation is the point with the lowest value on the vertical axis. This occurs at 2002.Choice B, C, and D are incorrect. The years 2004, 2005, and 2007 each had a greater number of graduates who enrolled in college within 24 months of graduation than did the year 2002.

Choice A is correct. The given equation is in slope-intercept form, or, where m is the value of the slope of the line of best fit. Therefore, the slope of the line of best fit is 0.096. From the definition of slope, it follows that an increase of 1 in the x-value corresponds to an increase of 0.096 in the y-value. Therefore, the line of best fit predicts that for each year between 1940 and 2010, the minimum wage will increase by 0.096 dollar per hour.Choice B is incorrect and may result from using the y-coordinate of the y-intercept as the average increase, instead of the slope. Choice C is incorrect and may result from using the 10-year increments given on the x-axis to incorrectly interpret the slope of the line of best fit. Choice D is incorrect and may result from using the y-coordinate of the y-intercept as the average increase, instead of the slope, and from using the 10-year increments given on the x-axis to incorrectly interpret the slope of the line of best fit.

Choice B is correct. A line of best fit is shown in the scatterplot such that as the value of \(x\) increases, the value of \(y\) decreases. Thus, the slope of the line of best fit shown is -. The slope of a line passing through two points, \((x_1, y_1)\) and \((x_2, y_2)\), can be calculated as \(y_2 -y_1/x_2 -x_1\). The line of best fit shown passes approximately through the points \((1, 12)\) and \((11, 4)\). Substituting \((1, 12)\) and \((11, 4)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in \(y_2 -y_1/x_2 -x_1\) gives \(4 -12/11 -1\), which is equivalent to \(-8 tenths\), or \(-0.8\). Therefore, of the given choices, \(-0.8\) is closest to the slope of the line of best fit shown. Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. The line of best fit shown has a -slope, not a positive slope. Choice D is incorrect. The line of best fit shown has a -slope, not a positive slope.

The correct answer is \(5\). For the graph shown, \(x\) represents time, in minutes, and \(y\) represents temperature, in °s Celsius \((° C)\). Therefore, the average rate of change, in \(° C\) per minute, of the recorded temperature of the air in the chamber between two x-values is the difference in the corresponding y-values divided by the difference in the x-values. The graph shows that at \(x = 5\), the corresponding y-value is \(14\). The graph also shows that at \(x = 7\), the corresponding y-value is \(24\). It follows that the average rate of change, in \(° C\) per minute, from \(x = 5\) to \(x = 7\) is \(24 -14/7 -5\), which is equivalent to \(10/2\), or \(5\).

Choice C is correct. At, the value of is less than the value of, which is equivalent to . As the value of x increases, the value of remains less than the value of until, which is when the two values are equal:, which is equivalent to . Then, for, the value of is greater than the value of . So there is a constant, 3, such that when, then, but when, then .Choice A is incorrect because when . Choice B is incorrect because when . Choice D is incorrect because when and when .

Choice A is correct. An equation of a line of best fit for data set F can be written in the form \(y = a + b x\), where \(a\) is the y-coordinate of the y-intercept of the line of best fit and \(b\) is the slope. The line of best fit shown for data set E has a y-intercept at approximately \((0, 12)\). It's given that data set F is created by multiplying the y-coordinate of each data point from data set E by \(3.9\). It follows that a line of best fit for data set F has a y-intercept at approximately \((0, 12(3.9))\), or \((0, 46.8)\). Therefore, the value of \(a\) is approximately \(46.8\). The slope of a line that passes through points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated as \(y_2 -y_1/x_2 -x_1\). Since the line of best fit shown for data set E passes approximately through the point \((12, 30)\), it follows that a line of best fit for data set F passes approximately through the point \((12, 30(3.9))\), or \((12, 117)\). Substituting \((0, 46.8)\) and \((12, 117)\) for \((x_1, y_1)\) and \((x_2, y_2)\), respectively, in \(y_2 -y_1/x_2 -x_1\) yields \(117 -46.8/12 -0\), which is equivalent to \(70.2/12\), or \(5.85\). Therefore, the value of \(b\) is approximately \(5.85\), or approximately \(5.9\). Thus, \(y = 46.8 + 5.9 x\) could be an equation of a line of best fit for data set F. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect. This could be an equation of a line of best fit for data set E, not data set F.

The correct answer is \(1 half\). For the graph shown, \(x\) represents time, in seconds, and \(y\) represents momentum, in newton-seconds. Therefore, the average rate of change, in newton-seconds per second, in the momentum of the object between two x-values is the difference in the corresponding y-values divided by the difference in the x-values. The graph shows that at \(x = 2\), the corresponding y-value is \(6\). The graph also shows that at \(x = 6\), the corresponding y-value is \(8\). It follows that the average rate of change, in newton-seconds per second, from \(x = 2\) to \(x = 6\) is \(8 -6/6 -2\), which is equivalent to \(2 fourths\), or \(1 half\). Note that 1/2 and .5 are examples of ways to enter a correct answer.

Choice D is correct. An appropriate model should follow the trend of the data points and should have data points both above and below the model. The scatterplot shows that the data points have an increasing trend that is curved. Therefore, an appropriate model should be an increasing curve with data points both above and below the model. Of the given choices, only the model in choice D is an increasing curve with data points both above and below the model. Choice A is incorrect. Since the trend of the data points isn't linear, a line isn't the most appropriate model for the data. Choice B is incorrect. Since the trend of the data points is increasing and isn't linear, a decreasing line isn't the most appropriate model for the data. Choice C is incorrect. All the data points are below the model shown in this graph.

Choice C is correct. For the given line graph, the percent of cars for sale at a used car lot on a given day is represented on the vertical axis. The percent of cars for sale is the smallest when the height of the line graph is the lowest. The lowest height of the line graph occurs for cars with a model year of \(2014\). Choice A is incorrect and may result from conceptual errors. Choice B is incorrect and may result from conceptual errors. Choice D is incorrect and may result from conceptual errors.

Choice A is correct. It's given that an equation for the exponential model shown can be written as \(y = a(b)Superscript x\), where \(a\) and \(b\) are positive constants. For an exponential model written in this form, if the value of \(b\) is greater than \(0\) but less than \(1\), the model is decreasing. If the value of \(b\) is greater than \(1\), the model is increasing. The exponential model shown is decreasing. Therefore, the value of \(b\) is greater than \(0\) but less than \(1\). Of the given choices, only \(0.83\) is a value greater than \(0\) but less than \(1\). Thus, \(0.83\) is closest to the value of \(b\). Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice B is correct. The line of best fit passes through the point . Therefore, the line of best fit predicts a temperature of 39°F for a location in Lake Tahoe Basin with an elevation of 8,500 feet.Choice A is incorrect. This is the lowest temperature listed on the scatterplot, and the line of best fit never crosses this value for any of the elevations shown. Choice C is incorrect. According to the line of best fit, the temperature of 41°F is predicted for an elevation of slightly greater than 7,500 feet, not an elevation of 8,500 feet. Choice D is incorrect. According to the line of best fit, the temperature of 43°F is predicted for an elevation of roughly 6,700 feet, not an elevation of 8,500 feet.

Choice B is correct. The line of best fit shown intersects the y-axis at a positive y-value and has a -slope. The graph of an equation of the form \(y = a + b x\), where \(a\) and \(b\) are constants, intersects the y-axis at a y-value of \(a\) and has a slope of \(b\). Of the given choices, only choice B represents a line that intersects the y-axis at a positive y-value, \(13.5\), and has a -slope, \(-0.8\). Choice A is incorrect. This equation represents a line that has a positive slope, not a -slope. Choice C is incorrect. This equation represents a line that intersects the y-axis at a -y-value, not a positive y-value, and has a positive slope, not a -slope. Choice D is incorrect. This equation represents a line that intersects the y-axis at a -y-value, not a positive y-value.

Choice A is correct. The average rate of change in temperature of the coffee in °s Fahrenheit per minute is calculated by dividing the difference between two recorded temperatures by the number of minutes in the corresponding interval of time. Since the time intervals given are all 10 minutes, the average rate of change is greatest for the points with the greatest difference in temperature. Of the choices, the greatest difference in temperature occurs between 0 and 10 minutes.Choices B, C, and D are incorrect and may result from misinterpreting the average rate of change from the graph.

Choice B is correct. The graphical model that most closely fits the data in the scatterplot is a model in which the number of data points above and below the model are approximately balanced. Fitting a graphical model to the data shown results in an upward-facing parabola with a y-intercept near and a vertex with an approximate x-value of 2.5. Of the given choices, only choice B gives an equation of an upward-facing parabola with a y-intercept at . Furthermore, substituting 2.5 for x into the equation in choice B yields . This is approximately the y-value of the vertex of the model.Choices A, C, and D are incorrect. These equations don’t give a graphical model that best fits the data. At, they have y-values of,, and 3, respectively. At, they have y-values of,, and 3, respectively.

Choice A is correct. The data point with has a y-coordinate of 12. The y-value predicted by the line of best fit at is approximately 11. The difference between the y-coordinate of the data point and the y-value predicted by the line of best fit at is, or 1.Choices B and C are incorrect and may result from incorrectly reading the scatterplot. Choice D is incorrect. This is the y-coordinate of the data point at .