Question

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

Answer 1

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

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 $(x_{1},y_{1})$ will be (0,0). We know for our problem that after 20 month he saved 1000, so our second point $(x_{2},y_{2})$ will be (20,1000).

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

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

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 $f(x)$ by $x$:
$f(x)_{AV}= \frac{f(x)}{x}$
$f(x)_{AV}= \frac{50x}{x}$
$f(x)_{AV}=50$

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