It is an extremely useful skill to be able to define functions from the information given in a word problem. But, it’s also something most people find difficult.

First, you’ll learn how to translate word problems into linear functions, $f(x)=ax+b$. In many cases, $a$ will represent the price per unit, and $b$ will represent a fixed price.

Rule

$a$ is the amount that varies (the unit price),

$b$ is the amount that occurs once (the fixed price).

Example 1

**Find a function for the price of base jump insurance with a fixed price of $\text{\$}\text{}1000\text{}$, and an additional price of $\text{\$}\text{}230\text{}$ per jump. **

The word problem has indicated that there is a fixed cost to you, no matter how many times you jump, in addition to an amount that changes with each jump you take. If you set $x$ equal to the number of jumps, the function looks like this:

$$f(x)=230x+1000$$ |

$230$ times the number of jumps, plus the fixed price of 1000.

Example 2

**Let’s say you have earned your driver’s license, and you are going on a trip and need to rent a car. You consider two types of rental agreements. The rental agreement “Smart Young 1” has a fixed payment of $\text{\$}\text{}25\text{}$ and then charges $\text{\$}\text{}7\text{}$ per 10 miles you drive the car. The rental agreement “Smart Young 2” has a fixed payment of $\text{\$}\text{}30\text{}$ and then $\text{\$}\text{}5\text{}$ per 10 miles you drive the car. Which rental agreement is most favorable if you will be driving 60 miles? **

In this case, you need to a function that represents the cost of each rental agreement, and plot them into the same coordinate system. Then, you need to make an assessment. In this case, to make the graph easier to read, you let $x$ represent 10 miles, so that $x=1$ is 10 miles, $x=2$ is 20 miles, etc.

The fixed price is $$25$, and the variable price is $$7$. That gives you the function

$$f(x)=7x+25$$ |

The fixed price is $$30$, and the variable price is $$5$. That gives you the function

$$g(x)=5x+30$$ |

Inserting the functions into a drawing tool produces these graphs:

From the graphs, you can see that rental agreement “Smart Young 2” is the cheaper option if you’re driving 60 miles. Rental agreement “Smart Young 1” is cheaper if you’re driving less than 25 miles.