in 2009 the number of vehicle sales in the untied states was 10,002 thousand and in 2003 it was 17,884 thousand
a)determine the average rate of change (slope) from 2009 to 2013
b) If x is the number of years since 2009 and f(x) is the number of vehicles sold, find the equation of the line through these two points
c) assuming f(x) is linear function, use the equation to predict the number of vehicles sold in 2016

The average rate of change in basically the slope of the line. We would need 2 points to find slope. THe slope is change in y points divided by change in x points.

Let 2009 be the initial year, so x = 0 at this point and each successive year is x = 1, x= 2, etc.

So, first point would be:

(0, 10002)

Second point therefore:

(4, 17884) [2013 is 4 years from 2009]

Now, lets find the Slope:

Slope = (17884 - 10002) / (4 - 0) = 1970.5 thousands per year = 1,970,500 per year

b)

Now, we have to find the equation of the line given the 2 points.

This is given by:

[tex]y-y_1=m(x-x_1)[/tex]

Where m is the slope found and x_1 would be first x point, which is 0 and y_1 would be first y point, which is 10002

SO,

[tex]y-10,002=1970.5(x-0)\\y-10,002=1970.5x\\y=1970.5x+10,002[/tex]

We change "y" to "f(x)" in the notation according to problem. Thus, the equation is:

[tex]f(x)=1970.5x+10,002[/tex] (in thousands)

c)

We have to predict the number of cars sold on 2016, which is 7 years from base year 2009. So,

x = 7

We will number of cars, that is f(x).

So we plug in x = 7 into the equation and find the value [f(7) basically]

So, we have:

[tex]f(x)=1970.5x+10,002\\f((7))=1970.5(7)+10,002\\f(7)=23,795.5[/tex]

So, number of cars in 2016 : 23,795,500