Find the rule of a piecewise linear function f(x) that models this data, that is a piecewise function with f(0) = 85.4, f(10)=96.1 and f(20) = 102.1

Question

The population f(x) of one city (in thousands) was 85.4 in 1970, 96.1 in 1980, and 102.1 in 1990. Let x=0 correspond to 1970 and x=20 to 1990. Find the rule of a piecewise linear function f(x) that models this data, that is a piecewise function with f(0) = 85.4, f(10)=96.1 and f(20) = 102.1 .

Answer 1

The phrase that sticks out in this problem is "piecewise linear function." If you understand that phrase, you should be able to solve the problem. So let's make sure we understand it.

 

"Piecewise" means it has more than one piece to it, meaning we'll have at least two sub-functions that make up this function as a whole. An example of this would be:

 

y = { x+1, if x < 0

         x-1, if x ≥ 0

 

In this function, for any x less than 0 we use the first sub-function, but for any x greater than or equal to 0 we use the second one.

 

"Linear" just means that the function makes a straight line when graphed. Any function that only has first-order variables is linear. A first-order variable is not raised to a power, so it's in a form like x, as opposed to x², or x³. Any function in slope-intercept form (f(x) = mx + b) is linear.

 

So the problem gives us three points: (0, 85.4), (10, 96.1), and (20, 102.1). Now we need lines to connect them. These lines will be linear functions, which we can use as sub-functions of our piecewise function.

 

The first line will be from (0, 85.4) to (10, 96.1). Let's get the slope:

m = (96.1 - 85.4) / (10 - 0)

= 10.7 / 10

= 1.07

We know the y-intercept is 85.4, so we are ready to make a function in slope-intercept form:

f(x) = 1.07x + 85.4

This sub-function will be true from x = 0 to x = 10.

 

The second line will be from (10, 96.1) to (20, 102.1). Let's start with the slope:

m = (102.1 - 96.1) / (20 - 10)

= 6 / 10

= .6

Now let's make a function in slope-intercept form:

f(x) = .6x + b

Let's solve for b using the point (10, 96.1):

96.1 = .6 (10) + b

96.1 = 6 + b

b = 90.1

Now we can put the y-intercept b back into the function:

f(x) = .6x + 90.1

This sub-function will be true from x = 10 to x = 20

 

Lastly, let's put the two sub-functions together into one piecewise function:

 

f(x) = { 1.07x + 85.4, if 0 ≤ x ≤ 10

             .6x + 90.1, if 10 < x ≤ 20

 

Sorry for the long explanation, but I hope it all makes sense!