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How Are Linear Functions Used in Real Life?

Applying mathematics to create functions from a practical problem is one of the most useful skills you can learn. This is a widely used skill in the field of mathematics, but also quite difficult for many students. When you translate real-world problems into math, you often formulate them as functions.

You will first learn how to translate a practical problem into linear functions, $f (x) = a x + b$ . Often, $a$ represents a unit price, while $b$ represents a fixed price.

Example 1

Find an expression for the price of a BASE jump insurance policy with a fixed price of $$ 1000$ and a price of $$ 230$ per jump

From the text you can see that there is an amount that is fixed regardless of how many jumps you do, and an amount that varies with the number of jumps. If you set $x$ equal to the number of jumps, the function looks like this:

f (x) = 230 x + 1000 .

The price is 230 times the number of jumps plus the fixed price.

Rule

How to Find a Linear Function

$a$ is the amount that varies,

$b$ is the amount that occurs one time.

Example 2

Let’s say you are old enough to have your driver’s license. You are going on a trip and have to rent a car. You are considering two types of leases.

Lease agreement “Smart Young 1” is such that you pay a fixed amount of $$ 24.90$ , and then $$ 7$ per 10 miles you drive the car.

Lease agreement “Smart Young 2” is such that you pay a fixed amount of $$ 29.90$ , and then $$ 5$ per 10 miles you drive. Which lease should you choose if you plan to drive a trip of 60 miles?

In this case, you need to find a function for each lease, then plot them in the same coordinate system. Then you need to make an assessment. In this case, let $x$ represent 10 miles, so that $x = 1$ is 10 miles, $x = 2$ is 20 miles, and so on.

Lease “Smart Young 1”

The fixed price is $ $25.00$ , and the cost per mile is $ $7$ . This gives you the function

f (x) = 7 x + 25

Lease “Smart Young 2”

The fixed price is $ $30.00$ , and the cost per mile is $ $5$ . This gives you the function

g (x) = 5 x + 30

You plot $f$ and $g$ and get these graphs:

Example of functions in real life

You can see that the two lines intersect at

x

2.5

. Remember that

x

represents 10 miles.

That means that lease “Smart Young 2” is cheaper if you drive over 25 miles, and “Smart Young 1” is cheaper if you drive less than 25 miles. They cost the same if you drive exactly 25 miles.

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