Modeling the Real World with Families of Functions

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Families of functions are exactly as their name indicates, groups of functions that are all related in a specific way. Let's look at families of functions and use examples to see how to use them to model and analyze real-world phenomena.

Families of Functions

You may be familiar with the fact that when someone tosses a ball up in the air, the ball's height off the ground can be represented using a quadratic function. For instance, suppose Tommy throws a ball up in the air.


Notice that the ball's height off the ground increases, hits a maximum, then decreases. This shape is called a parabola, and it is the shape of the graph of a quadratic function.

Now, in Tommy's case, the height of the ball can be represented with the function:

h(x) = -16x2 + 64x + 5

where x is the number of seconds after the ball leaves Tommy's hands. (Since the ball is starting in Tommy's hand it starts about five feet off the ground.) But, what would happen if Tommy threw the ball up in the air while standing on a balcony that is 10 feet off the ground? If this is the case, the function representing the ball's height is:

f(x) = -16x2 + 64x + 15

Hmm, so what would happen if Tommy threw the ball a little harder or softer initially, or if he threw the ball at a different angle initially?


In each case no matter how we vary the conditions, the function is still quadratic. It is just the constants and coefficients that are changed. Here's an interesting fact! The group of different quadratic functions that can represent the ball's height make up a family of functions.

A family of functions is a group of functions that can all be derived from transforming a single function called the parent function. The parent function is the most basic function in the family of functions, the function from which all the other functions in the family can be derived.

In this instance, the family of functions is the quadratic function and the parent function is:

f(x) = x2

Notice that if we take f(x) = x2 and shift it right horizontally, stretch it vertically, reflect it over the x-axis, and shift it up vertically, we end up with Tommy's initial function.


Any function in a family of functions can be derived from the parent function by taking it through some transformations. Knowing what type of transformation to use is another lesson, but we can get an idea of how useful families of functions are!

Real World Examples

Now that we have derived Tommy's function from the parent function, we can use it to analyze the situation. For instance, we can determine when the ball will hit the ground by setting the function equal to zero (when the height off the ground is zero) and solving for x.


We see that the ball hits the ground just after 4 seconds. See how useful these families of functions are when it comes to modeling and analyzing the real world?

Pretty neat so far, huh? Well, what's even more interesting is that there are plenty more instances in which families of functions can be used to model the world around us. We just saw that we can use the quadratic family of functions to represent a ball being tossed in the air. The function is dependent on the varying conditions of the scenario, but it is always a quadratic function.

In general, families of functions are used to model real-world phenomena in which conditions vary, so the type of function used to represent the phenomena all come from the same function family, but they vary within the family based on the conditions.

As another example, consider something as simple as how long a vitamin is detectable our body after we eat it. The rate at which a vitamin leaves our body can be modeled with the family of exponential functions.

An exponential function is one in which the variable is in the exponent.


We see that vitamins leave the body quickly at first and then more slowly. In general, this type of pattern is modeled by the function:

f(x) = abx + c

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