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Describe correlation EXAMPLE 1 Telephones Describe the correlation shown by each scatter plot.

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Describe correlation EXAMPLE 1 SOLUTION The first scatter plot shows a positive correlation, because as the number of cellular phone subscribers increased, the number of cellular service regions tended to increase. The second scatter plot shows a negative correlation, because as the number of cellular phone subscribers increased, corded phone sales tended to decrease.

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EXAMPLE 2 Estimate correlation coefficients Tell whether the correlation coefficient for the data is closest to – 1, – 0.5, 0, 0.5, or 1. a. SOLUTION a. The scatter plot shows a clear but fairly weak negative correlation. So, r is between 0 and – 1, but not too close to either one. The best estimate given is r = – 0.5. (The actual value is r – 0.46.)

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Estimate correlation coefficients EXAMPLE 2 b. The scatter plot shows approximately no correlation. So, the best estimate given is r = 0. (The actual value is r – 0.02.) SOLUTION b.

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Estimate correlation coefficients EXAMPLE 2 c. c. The scatter plot shows a strong positive correlation. So, the best estimate given is r = 1. (The actual value is r 0.98.) SOLUTION

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EXAMPLE 3 Approximate a best-fitting line The table shows the number y (in thousands) of alternative-fueled vehicles in use in the United States x years after 1997. Approximate the best-fitting line for the data. x01234567 y280295322395425471511548 Alternative-fueled Vehicles

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Approximate a best-fitting line EXAMPLE 3 SOLUTION STEP 1 Draw a scatter plot of the data. STEP 2 Sketch the line that appears to best fit the data. One possibility is shown.

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Approximate a best-fitting line EXAMPLE 3 STEP 3 Choose two points that appear to lie on the line. For the line shown, you might choose (1, 300), which is not an original data point, and (7, 548), which is an original data point. STEP 4 Write an equation of the line. First find the slope using the points (1, 300) and (7, 548). 248 6 m = = 41.3 548 – 300 7 – 1

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Approximate a best-fitting line EXAMPLE 3 Use point-slope form to write the equation. Choose (x 1, y 1 ) = (1, 300). y – y 1 = m(x – x 1 ) Point-slope form y – 300 = 41.3(x – 1) Substitute for m, x 1, and y 1. Simplify. y 41.3x + 259 ANSWER An approximation of the best-fitting line is y = 41.3x + 259.

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EXAMPLE 4 Use a line of fit to make a prediction Use the equation of the line of fit from Example 3 to predict the number of alternative-fueled vehicles in use in the United States in 2010. SOLUTION Because 2010 is 13 years after 1997, substitute 13 for x in the equation from Example 3. y = 41.3x + 259 = 41.3(13) + 259 796

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Use a line of fit to make a prediction EXAMPLE 4 ANSWER You can predict that there will be about 796,000 alternative-fueled vehicles in use in the United States in 2010.

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Use a graphing calculator to find a best-fitting line EXAMPLE 5 Use the linear regression feature on a graphing calculator to find an equation of the best-fitting line for the data in Example 3. SOLUTION STEP 1 Enter the data into two lists. Press and then select Edit. Enter years since 1997 in L 1 and number of alternative-fueled vehicles in L 2.

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Use a graphing calculator to find a best-fitting line EXAMPLE 5 STEP 2 Find an equation of the best- fitting (linear regression) line. Press choose the CALC menu, and select LinReg(ax + b). The equation can be rounded to y = 40.9x + 263.

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Use a graphing calculator to find a best-fitting line EXAMPLE 5 STEP 3 Make a scatter plot of the data pairs to see how well the regression equation models the data. Press [STAT PLOT] to set up your plot. Then select an appropriate window for the graph.

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Use a graphing calculator to find a best-fitting line EXAMPLE 5 STEP 4 Graph the regression equation with the scatter plot by entering the equation y = 40.9x + 263. The graph (displayed in the window 0 ≤ x ≤ 8 and 200 ≤ y ≤ 600) shows that the line fits the data well. An equation of the best-fitting line is y = 40.9x + 263. ANSWER

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EXAMPLE 1 Graph a function of the form y = | x – h | + k Graph y = | x + 4 | – 2. Compare the graph with the graph of y = | x |. SOLUTION STEP 1 Identify and plot the vertex, (h, k) = (– 4, – 2). STEP 2 Plot another point on the graph, such as (–2, 0). Use symmetry to plot a third point, (– 6, 0 ).

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Graph a function of the form y = | x – h | + k EXAMPLE 1 STEP 3 Connect the points with a V-shaped graph. STEP 4 Compare with y = | x |. The graph of y = | x + 4 | – 2 is the graph of y = | x | translated down 2 units and left 4 units.

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Graph functions of the form y = a | x | EXAMPLE 2 Graph (a) y = | x | and (b) y = – 3 | x |. Compare each graph with the graph of y = | x |. 1 2 SOLUTION a.The graph of y = | x | is the graph of y = | x | vertically shrunk by a factor of. The graph has vertex (0, 0) and passes through (– 4, 2) and (4, 2). 1 2 1 2

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Graph functions of the form y = a | x | EXAMPLE 2 b.The graph of y = – 3 | x | is the graph of y = | x | vertically stretched by a factor of 3 and then reflected in the x -axis. The graph has vertex (0, 0) and passes through (– 1, – 3) and (1, – 3).

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EXAMPLE 3 Graph a function of the form y = a x – h + k Graph y = –2 x – 1 + 3. Compare the graph with the graph of y = x. SOLUTION Identify and plot the vertex, (h, k) = (1, 3). Plot another point on the graph, such as (0, 1). Use symmetry to plot a third point, (2, 1). STEP 1 STEP 2

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EXAMPLE 3 Graph a function of the form y = a x – h + k STEP 3 Connect the points with a V- shaped graph. Compare with y = x. The graph of y = – 2 x – 1 + 3 is the graph of y = x stretched vertically by a factor of 2, then reflected in the x- axis, and finally translated right 1 unit and up 3 units. STEP 4

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Apply transformations to a graph EXAMPLE 5 The graph of a function y = f (x) is shown. Sketch the graph of the given function. a. y = 2 f (x) b. y = – f (x + 2) + 1

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Apply transformations to a graph EXAMPLE 5 The graph of y = 2 f (x) is the graph of y = f (x) stretched vertically by a factor of 2. (There is no reflection or translation.) To draw the graph, multiply the y-coordinate of each labeled point on the graph of y = f (x) by 2 and connect their images. a. SOLUTION

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Apply transformations to a graph EXAMPLE 5 The graph of y = – f (x + 2) +1 is the graph of y = f (x) reflected in the x-axis, then translated left 2 units and up 1 unit. To draw the graph, first reflect the labeled points and connect their images. Then translate and connect these points to form the final image. b. SOLUTION

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