Question

find a quadratic equation that models the decline in physical recorded music revenue (not digital). let x = number of years since 2000 and y be the physical recorded music revenue in millions of dollars. use this model to fill in the predicted physical revenue below and compare with the actual given data.

year	actual physical music revenue (in millions of dollars)
2009	4440
2010	3649
2011	3112
2012	2669
2013	2317
2014	2047

use the model in the previous step to fill in the predicted physical revenue below and compare with the actual given data. (round to the nearest integer as needed)

x	actual revenue	predicted revenue (in millions of dollars)	difference between actual and predicted
2009	4440	4515	-75
2010	3649	3723	-74
2011	3112	3086	26
2012	2669	□	□

Explanation:

Step1: Define x values for each year

For 2008: $x=8$; 2009: $x=9$; 2010: $x=10$; 2011: $x=11$; 2012: $x=12$; 2013: $x=13$; 2014: $x=14$

Step2: Derive quadratic model from data

Using the given actual data, the quadratic regression model is:
$$y = -4.87x^2 - 192.54x + 7284.76$$

Step3: Calculate 2012 predicted revenue

Substitute $x=12$ into the model:
$$y = -4.87(12)^2 - 192.54(12) + 7284.76$$
$$y = -4.87(144) - 2310.48 + 7284.76$$
$$y = -701.28 - 2310.48 + 7284.76 = 4273.00 \approx 4273$$

Step4: Calculate 2012 difference

Subtract predicted from actual revenue:
$$2669 - 4273 = -1604$$

Step5: Calculate 2013 predicted revenue

Substitute $x=13$ into the model:
$$y = -4.87(13)^2 - 192.54(13) + 7284.76$$
$$y = -4.87(169) - 2503.02 + 7284.76$$
$$y = -823.03 - 2503.02 + 7284.76 = 3958.73 \approx 3959$$

Step6: Calculate 2013 difference

Subtract predicted from actual revenue:
$$2317 - 3959 = -1642$$

Step7: Calculate 2014 predicted revenue

Substitute $x=14$ into the model:
$$y = -4.87(14)^2 - 192.54(14) + 7284.76$$
$$y = -4.87(196) - 2695.56 + 7284.76$$
$$y = -954.52 - 2695.56 + 7284.76 = 3634.68 \approx 3635$$

Step8: Calculate 2014 difference

Subtract predicted from actual revenue:
$$2047 - 3635 = -1588$$

Answer:

Predicted Revenue:

2012: 4273, 2013: 3959, 2014: 3635

Difference between Actual and Predicted:

2012: -1604, 2013: -1642, 2014: -1588

Quadratic Model:

$y = -4.87x^2 - 192.54x + 7284.76$