We want to determine the functions relating the year to the number of viewers of two shows are linear or nonlinear. Let's find the equation that represents this relationship for each show.
Show A
We are told that in their first year, Show A has 7 million viewers. Each year, the show keeps 90 percent of the viewers it had in the previous years. Consider that the total viewers for Show A in any given year will be equal to the product of 0.9 and the number of previous viewers. Here, 0.9 represents the 90 percent of remaining viewers and x the year after the first.
Year, x
Change in Viewers
Function
Viewership, y (millions)
1
7
7 * 0.9^(1-1)=7* 0.9^0
7
2
7 * 0.9
7 * 0.9^(2-1)=7* 0.9^1
6.3
3
7 * 0.9 * 0.9
7 * 0.9^(3-1)=7* 0.9^2
5.67
4
7 * 0.9 * 0.9 * 0.9
7 * 0.9^(4-1)=7* 0.9^3
5.103
5
7 * 0.9 * 0.9 * 0.9 * 0.9
7 * 0.9^(5-1)=7* 0.9^4
4.5927
6
7 * 0.9 * 0.9 * 0.9 * 0.9 * 0.9
7 * 0.9^(6-1)=7* 0.9^5
4.13343
Recall that the graph of a linear function shows a constant rate of change and is always a straight line. A nonlinear function does not have a constant rate of change, so its graph is not a line. To determine which type the function for Show A's viewership is, let's plot the ordered pairs we found in our table and try to connect them with a line.
We cannot connect all the dots with a straight line, which suggests that this function is nonlinear.
Show B
The Show B has 5 million viewers in the first year but it loses 200 000 viewers each year. This means that the total viewers for Show B will be equal to the difference between the viewers of the previous year and 200 000. Let's make a table like we did in Part A.
Year, x
Change in Viewers
Function
Viewership, y (millions)
1
5
5 -0.2* ( 1-1)=5
5
2
5-0.2
5 -0.2* ( 2-1)=5-0.2
4.8
3
5 -0.2-0.2
5 -0.2* ( 3-1)=5-0.4
4.6
4
5 -0.2-0.2-0.2
5 -0.2* ( 4-1)=5-0.6
4.4
5
5 -0.2-0.2-0.2-0.2
5 -0.2* ( 5-1)=5-0.8
4.2
6
5 -0.2-0.2-0.2-0.2-0.2
5 -0.2* ( 6-1)=5-1
4
Now let's plot these points and see what the graph looks like.
We can connect all the points along a single straight line, which means that this function is linear.
In Part A, we found the numbers of viewers for two shows after six years.
Show A
Show B
4 133 430
4 000 000
From the table we can see that the Show A has more viewers in its sixth year than Show B does.
