It takes Matt 20 months to save 1,000. Write an equation that models the average number of dollars, x, Matt saves each month.

To solve this we are going to use the slope formula: [tex]m= \frac{y_{2}-y_{1}}{x_{2}-x_{1} } [/tex], and the point slope formula: [tex]y-y_{1}=m(x-x_{1})[/tex]

For our problem we can infer that prior to the 20 months period of saving, he didn't have any savings at all. So our first point [tex](x_{1},y_{1})[/tex] will be (0,0). We know for our problem that after 20 month he saved 1000, so our second point [tex](x_{2},y_{2})[/tex] will be (20,1000).

Now that we have our points, we can use our slope formula to find [tex]m[/tex]:
[tex]m= \frac{y_{2}-y_{1}}{x_{2}-x_{1} } [/tex]
[tex]m= \frac{1000-0}{20-0} [/tex]
[tex]m= \frac{1000}{20} [/tex]
[tex]m=50[/tex]

Now that we have our slope, we can use our point slope formula:
[tex]y-y_{1}=m(x-x_{1})[/tex]
[tex]y-0=50x-0[/tex]
[tex]y=50x[/tex]
Remember that [tex]y[/tex] and [tex]f(x)[/tex] are equivalent, so:
[tex]f(x)=50x[/tex]

Finally, to find the average number of dollars Matt saves each month, we are going to find the average function. To do that we are going to divide our function [tex]f(x)[/tex] by [tex]x[/tex]:
[tex]f(x)_{AV}= \frac{f(x)}{x} [/tex]
[tex]f(x)_{AV}= \frac{50x}{x} [/tex]
[tex]f(x)_{AV}=50[/tex]

We can conclude that the function that models the average number of dollars, x, Matt saves each month is [tex]f(x)_{AV}=50[/tex]