The director of the town's sports programs is studying trends in enrollment over time. She finds that the number of participants in the town's soccer program increases by 10% each year, and the number of participants in the town's street hockey program increases by 10 children each year. This year, there are 120 children participating in each program. Which function can be used to predict the total number of children participating in the town's soccer and street hockey programs in x years?

Suppose, after [tex]x[/tex] years, the number of participants in the town's soccer program is [tex]y_{1}[/tex] and in the town's street hockey program is [tex]y_{2}[/tex]

This year, there are 120 children participating in each program.

The number of participants in the town's soccer program increases by 10% each year.

Using the exponential growth formula, the number of participants in the town's soccer program after [tex]x[/tex] years will be.....

[tex]y_{1}=120(1+0.10)^x\\ \\ y_{1}=120(1.10)^x[/tex]

The number of participants in the town's street hockey program increases by 10 children each year.

So, the number of increased participants in [tex]x[/tex] years [tex]=10x[/tex]

Thus, the number of participants in the town's street hockey program after [tex]x[/tex] years will be.....

[tex]y_{2}=120+10x[/tex]

If the total number of children participating in the two programs is represented as [tex]y[/tex], then.....

[tex]y= y_{1}+y_{2}\\ \\ y=120(1.10)^x+120+10x[/tex]

So, the function that can be used to predict the total number of children participating in the town's soccer and street hockey programs in x years will be: [tex]y=120(1.10)^x+120+10x[/tex]