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If $f(x)$ is a linear function, with $f(2)=\u201311,$ and $f(4)=-25,$ find an equation for the function in slope-intercept form.
$y=-7x+3$
In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.
Given a linear function $f$ and the initial value and rate of change, evaluate $f\left(c\right).$
Marcus currently has 200 songs in his music collection. Every month, he adds 15 new songs. Write a formula for the number of songs, $N,$ in his collection as a function of time, $t,$ the number of months. How many songs will he own in a year?
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 in 1 year (12 months). In other words, we can evaluate the function at $t=12.$
Marcus will have 380 songs in 12 months.
Working as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilya’s weekly income, $I,$ depends on the number of new policies, $n,$ he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies and earned $920. Find an equation for $I\left(n\right),$ and interpret the meaning of the components of the equation.
The given information gives us two input-output pairs: $\left(\mathrm{3,760}\right)$ and $\left(\mathrm{5,920}\right).$ 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.
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