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McDougal Littell Algebra 1: Online Textbook Help13 chapters | 144 lessons

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will discuss how mathematical models are used to represent real world phenomena. We will look at linear, exponential, and quadratic models and their characteristics.

Imagine you are going for a drive. You start off 3 miles from your house, and you are traveling further away from your house at 30 mph enjoying the scenery. I'm going to show you something really neat! You can actually model how far you are from your house using the equation *y* = 30*x* + 3, where *y* is the distance you are from your house, and *x* is the number of hours you've been driving.

So, why is this neat? Well, because you can use this model to calculate your distance from your house at any time during your drive. For instance, suppose you've been driving for half an hour. You can plug 1/2 in for *x* and figure out how far you are from your house.

After half an hour of driving, you are 18 miles from your house.

The equation representing your distance is an example of a **mathematical model**. We use mathematical models to represent real world phenomena and solve problems all the time. We are going to look at three types of mathematical models -- linear, exponential, and quadratic -- and we are going to look at characteristics for each.

The equation *y* = 30*x* + 3 in our driving example is an example of a **linear model**. In general, a linear model takes on the following form:

In a linear equation, the highest exponent of the variable *x* is 1. The graph of a linear equation is a line. To see this, observe the graph of our linear model *y* = 30*x* + 3.

We see that the graph is a line. This tells us that the distance you are from your house is changing at a constant rate. We see that by modeling the situation with a linear model, as we did, we are able to better analyze the situation.

Consider the following scenario. You have $100 that you would like to put into a savings account. You go to the bank, and they say they will give you 2% interest compounded annually. Wouldn't it be great if you were able to know how much money would be in the account after any given number of years? I've got good news! Once again, we have a mathematical model that will solve this problem. That is, the equation *y* = 100(1 + .02) *x*, or *y* = 100(1.02) *x*, where *y* is the amount in the account after *x* years, gives us the desired information.

Awesome! We can know exactly how much will be in the account after a given number of years. For instance, if we wanted to know how much will be in the account after 2 years, we plug 2 in for *x* and evaluate.

We see that after 2 years, the account will have $104.04 in it. Congrats! You made $4.04 in interest. Not much, but every little bit helps!

This model is an example of an exponential model. In general, an **exponential model** takes on the following form:

A phenomenon that can be modeled by an exponential model takes on the pattern of increasing/decreasing slowly then quickly, or quickly then slowly. For instance, let's take a look at the graph of our exponential model *y* = 100(1.02) *x*.

We see that the graph is increasing slowly at first, and then more quickly, and it takes on a general shape of an exponential model. We can use our model and its graph to analyze your savings account.

Are you a fan of basketball? Even if you aren't, you have probably heard of a free throw. Well, guess what? This is yet another example of real world phenomena that can be modeled mathematically. In general, a free throw can be represented by a quadratic model. A **quadratic model** takes on the following general form:

We model phenomena with a quadratic model when it takes on the pattern of increasing, reaching a peak, then decreasing, or decreasing, reaching a low point, then increasing. To picture this, think of a free throw in basketball. The ball leaves the player's hands, goes up in the air, hits a high point, then begins to fall -- hopefully into the basket.

Because of how a quadratic model behaves, the graph of a quadratic equation takes on the shape of a U or an upside-down U. We call the graph of a quadratic equation a **parabola**, and the high or low point of the graph is a **vertex**.

To illustrate this, consider the free throw modeled by the quadratic model *y* = -0.06*x* 2 + 1.5*x* + 6. The graph of this model is shown.

See that the graph forms an upside-down U with a maximum point.

**Mathematical models** can be used to represent real world phenomena and allow us to better analyze problems. We've looked at **linear**, **exponential** and **quadratic** models. The general form of each model is shown:

Each model has different characteristics that make them appropriate to use to model different scenarios. If you are modeling something that is increasing or decreasing at a constant rate, a linear model is a good way to go. If you are modeling something that is increasing or decreasing slowly then quickly, or quickly then slowly, then you should try to use an exponential model. Lastly, if you are modeling something that increases, hits a max, then decreases, or decreases, hits a minimum, then increases, use a quadratic model. By being familiar with these different types of models and how they behave, we have a better idea of which ones to use in different situations.

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McDougal Littell Algebra 1: Online Textbook Help13 chapters | 144 lessons

- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- How to Solve Quadratics That Are Not in Standard Form 6:14
- Solving Quadratic Equations with Square Roots or Graphing the Function
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- How to Use the Quadratic Formula to Solve a Quadratic Equation 9:20
- How to Solve Quadratics with Complex Numbers as the Solution 5:59
- Comparing Linear, Exponential & Quadratic Models
- Go to McDougal Littell Algebra 1 Chapter 10: Quadratic Equations & Functions

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