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Common Core Math Grade 8 - Functions: Standards5 chapters | 19 lessons

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Lesson Transcript

Instructor:
*John Sepanski*

John has taught 6th Grade Mathematics through Geometry and has a Master's degree in Education

In this lesson, you will learn to determine the initial value of a function by studying functions in the form of word problems, graphs and linear equations.

Almost everybody today has a cell phone. Some people have very basic phones; others have their brothers' and sisters' old phones; some people have really beat-up phones! Well, pretty soon, that one big computer company will be releasing the latest and greatest new phone and everybody's going to want one, especially me!

But in order to afford that new phone, I'm going to have to sign a contract. A contract is a document that says you can get the new phone for a special price as long as you agree to stay with the cellular company for two years and make monthly payments for your data usage, texts and calls. This is a great example of the initial value of a function.

The **initial value of a function** is the point at which a function begins. Well, that makes sense, but what does it really mean? A **function** is a mathematical relation into which we input values of a domain that generate output values of a range. Let's take a step back to our phone example to help break this concept down a little.

Suppose that when it's released, the cost of the latest, greatest, coolest phone is $650, but you can get it for only $200 as long as you're willing to sign a contract. That contract is going to include your agreement to a two-year agreement with a plan including voice, data and text. Hang on a minute. All those words mean that your phone isn't really going to cost you just $200. It means that you start by paying $200, but then you pay an amount, say $110, every month for two years.

Your initial value was just the beginning! Your phone actually will cost you $2,840 over the time of two years. I wonder if I can buy the phone without a contract? It will probably cost a lot more, but it might be worth it, but then I would have to pay for data anyways.

Let's take a look at some strictly mathematical functions. Consider the example *f*(*x*) = 2*x* + 10.

Maybe you could think of this standing for the cost of riding rides at a carnival. It costs $10 to get in, but it will cost you $2 more for every ride you get on. In this case, the initial value of the function is 10. In other words, if you ride 0 rides and just go to the fair, it will cost you $10.

We can go on and determine how much it will cost if you ride more rides. If you ride 1 ride, you will pay $12. If you ride 5 rides, you will pay a total of $20 because you multiply 5 times 2 and then add 10. If you ride 8 rides, you will pay a total of $26 because you multiply 8 times 2 and then add 10. You're always going to pay more than ten dollars because the initial cost of the function is 10: cost of admission = $10.

One more way to think of the initial value of a function is to evaluate the function when the domain = 0; that is, evaluate *f*(0) = 2*x* + 10. If I do that, when I substitute 0 in for *x*, 2(0) + 10 = 0 + 10, and everybody knows that that equals 10!

In order to look closer at our function, we can take the points from our table and plot them on a graph. You can immediately see the initial value of a function on a graph because it is where the line starts.

The **initial value of a function** is where a **function** starts. One way to think of initial value is to remember things like the cell phone contract or other items that you initiate by paying a certain amount.

Take this pizza menu, for example. It's just like the cell phone contract example that we used in the lesson. In this case, the medium pizza costs $11.99 and each additional topping costs $1.69. Word problems are converted to function notation: *f*(*t*) = 1.69*t* + 11.99. It might be easier to remember if we start with the cost of $11.99, and then add $1.69 for each additional topping to get the total cost of the pizza: 11.99 + 1.69*t* = *f*(*t*).

But mathematicians like to keep everything in order of operations, where multiplication comes before addition. Regardless of what you order on a pizza, you're going to pay at least $11.99 because that is the initial value of a pizza. Mathematicians sure have a way of making pizza seem much more complicated than it is! And that's why you don't see a lot of mathematicians cooking pizzas. And you sure don't see a lot of mathematicians selling cell phones!

The purpose of this lesson is to help you to:

- Provide the definition of a function
- Think of initial values in real-life terms
- Work through related examples

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Common Core Math Grade 8 - Functions: Standards5 chapters | 19 lessons

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