4.1 Linear functions  (Page 6/27)

The initial value for this function is 200 because he currently owns 200 songs, so $N(0)=200,$ which means that $b=200.$

The number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. Therefore we know that $m=15.$ We can substitute the initial value and the rate of change into the slope-intercept form of a line.

We can write the formula $N(t)=15t+200.$

With this formula, we can then predict how many songs Marcus will have at the end of one year (12 months). In other words, we can evaluate the function at $t=12.$

Marcus will have 380 songs in 12 months.

The given information gives us two input-output pairs: $(3,760)$ and $(5,\text{92}0).$ We start by finding the rate of change.

Keeping track of units can help us interpret this quantity. Income increased by $160 when the number of policies increased by 2, so the rate of change is $80 per policy. Therefore, Ilya earns a commission of $80 for each policy sold during the week.

We can then solve for the initial value.

The value of $b$ is the starting value for the function and represents Ilya’s income when $n=0,$ or when no new policies are sold. We can interpret this as Ilya’s base salary for the week, which does not depend upon the number of policies sold.

We can now write the final equation.

Our final interpretation is that Ilya’s base salary is $520 per week and he earns an additional $80 commission for each policy sold.

We can see from the table that the initial value for the number of rats is 1000, so $b=1000.$

Rather than solving for $m,$ we can tell from looking at the table that the population increases by 80 for every 2 weeks that pass. This means that the rate of change is 80 rats per 2 weeks, which can be simplified to 40 rats per week.

If we did not notice the rate of change from the table we could still solve for the slope using any two points from the table. For example, using $(2,1080)$ and $(6,1240)$