Create a linear function that models the following real-world financial problem:
Ray puts $6,000 into a savings account. Every month, he adds $200 to the account. Write a function [tex]f(x)[/tex] that models the total amount in dollars he puts into the savings account after [tex]x[/tex] months.
A. [tex]f(x)=6,000 x+200[/tex]
B. [tex]f(x)=6,000 x-200[/tex]
C. [tex]f(x)=200 x+6,000[/tex]
D. [tex]f(x)=200 x-6,000[/tex]
Answer (1)
The problem describes a savings account with an initial deposit and monthly additions.
The linear function f ( x ) models the total amount in the account after x months.
The slope of the function is the monthly addition ($200), and the y-intercept is the initial deposit ($6,000).
The function is f ( x ) = 200 x + 6000 , so the answer is f ( x ) = 200 x + 6 , 000 .
Explanation
Problem Analysis Let's analyze the problem. Ray starts with an initial deposit of $6,000 in his savings account. Every month, he adds an additional 200 t o t h e a cco u n t . W e n ee d t ocre a t e a l in e a r f u n c t i o n , d e n o t e d a s f(x) , t ha t re p rese n t s t h e t o t a l am o u n t o f m o n ey in t h e a cco u n t a f t er x$ months.
Identifying Slope and Intercept The function f ( x ) will be in the form of a linear equation: f ( x ) = m x + b , where m is the slope (the rate of change) and b is the y-intercept (the initial value). In this case, the rate of change is the amount added each month, which is $200. The initial value is the initial deposit, which is $6,000.
Formulating the Function Therefore, the function that models the total amount in the savings account after x months is: f ( x ) = 200 x + 6000
Final Answer The correct answer is f ( x ) = 200 x + 6 , 000 .
Examples
Imagine you're saving up for a new bicycle that costs $1,000. You start with $200 in your piggy bank and decide to add 50 e a c h w ee k f ro m yo u r a ll o w an ce . U s in g a l in e a r f u n c t i o n , yo u c an e a s i l yc a l c u l a t e h o w man y w ee k s i tw i llt ak e t os a v ee n o ug hm o n ey . T h e f u n c t i o n w o u l d b e f(x) = 50x + 200 , w h ere x i s t h e n u mb ero f w ee k s . B yse tt in g f(x)$ to 1 , 000 an d so l v in g f or x$, you can determine the number of weeks needed to reach your goal. This kind of problem helps in planning savings and understanding how consistent contributions lead to achieving financial goals.
