Using Quadratic Functions to Model a Given Data Set or Situation

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  • 0:01 Quadratic Functions
  • 0:59 Modeling
  • 1:56 A Data Set
  • 3:38 A Situation
  • 4:49 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how quadratic functions are used to model situations and data gathered from real world scenarios. See how our function fits our data and situation well.

Quadratic Functions

Quadratic functions are those functions with a degree of 2. What this means is that they will have, at most, three terms, and the highest exponent is always a 2. Yes, quadratic functions will always have the term with the exponent of 2.

The general or standard form of all quadratic functions is f(x) = ax^2 + bx + c, where a, b, and c are your coefficients, and x is your variable. Your coefficients can be any number. If a coefficient is 0, then it makes that term disappear. For quadratic functions, though, because the ax^2 term always needs to be present, the coefficient of a cannot be 0.

Your variable, x, can be any letter that is convenient for the function. If we are talking about time, our variable can be t. If we are talking about height, then our variable can be h. Our variable can be any letter that makes our function easy to understand.


Of course, you come across these functions in math problems and tests and exams. But these functions also appear in the real world as models of real life events. For example, and this is just one example, the path an object, such as a baseball, travels after its been hit follows this quadratic function:

quadratic modeling

Where g is 4.9 for meters and 16 for feet, v sub 0 is our initial velocity, and h sub 0 is our initial height. Our coefficient g comes from the force of gravity and has been calculated specific to Earth.

Our initial velocity is the speed the object is traveling upwards at when released, and our initial height is the height at which our object is released. Our function then gives the height of the object at any given time. Let's take a closer look at modeling using this quadratic function.

A Data Set

We can be given a data set, a collection of data points, to consider. Since we are dealing with a falling object after it is released, we will look at data points that follow the path of a falling object, such as a human cannonball. Say we are given the following data points:

time (s) height (m)
0 3
1 14.8
2 16.8
3 9

We are told that these data points follow the path of a human cannonball under the force of gravity on Earth. We already know what the quadratic function looks like. So, we just need to find the initial velocity, the initial height, and whether we are dealing in meters or feet.

We look at our data points and see that at time 0, our human cannonball was at a height of 3. That means that our initial height is 3 meters. Since we are dealing in meters, our g is 4.9. To find our initial velocity, then, we can use our second data point, (1, 14.8).

We plug in these values into our quadratic function for the movement of an object under gravity. We get 14.8 = - 4.9 * 1^2 + v sub 0 * 1 + 3. This turns into 14.8 = -4.9 + v sub 0 + 3. Adding like terms, we get 14.8 = -1.9 + v sub 0.

Solving for v sub 0 by adding 1.9 to both sides gives us an initial velocity of 16.7 meters per second. Okay, we have all the information we need to build our quadratic function. Plugging in all our necessary items, we get this for the quadratic function that models this set of data:

quadratic modeling

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