What’s my rule? 3 Input (x) output(y) ? 5 1 6 2 7 3

Today, we are going to learn how to generalize a variety of patterns to a symbolic rule from tables, graphs, or words ~ SPI 706.1.2

You are going to make some notations in your journals. Snoopy’s Journal

My Function Machine takes a value called input and performs one or more operations on it according to a “rule” to produce a new value called the output.

A function is a relation that assigns exactly one output value to one input value output(y) 1 2 You can organize the input-output values of a function in a function table. 5 6 7 3 8

The function rule describes the relationship between each input and output. (x) output(y) 1 2 5 6 Add 5 7 3 8

I can use an algebraic expression to represent the rule of my function machine 5 X + 5

You can also use a “rule” to describe a pattern in a sequence of numbers. For example: 8, 16, 24, 32, … A sequence is a list of numbers in a specific order. What rule would you use for the above sequence? +8

Add 8 Each number in a list is called a term of the sequence. An Arithmetic sequence can be found by adding the same number to the previous term. 8, 16, 24, 32, … +8 +8 +8

Times 5 In a Geometric Sequence, each term is found by multiplying the previous term by the same number. For example: 2, 10, 50, 250, … x 5 x 5 x 5

You can also write an algebraic expression to describe a sequence You can also write an algebraic expression to describe a sequence. The value of each term can be described as a function of its position in the sequence. X + 5

You can show a sequence in a table You can show a sequence in a table. The position of the number in the sequence is “x” and the value of each term is “y” Position (x) 1 2 3 4 Value (y) 5 7 9 11 Notice the value increases by 2, so the first part of the rule would be 2x

If the rule were simply 2x, then the value for position 1 would be 2, but its 3 more. That means you add 3 to 2x to get the next term in the sequence. Position 1 2 3 4 Value 5 7 9 11 Let’s check the rule 2x + 3! 2(1) + 3 = 5; 2(2) + 3 = 7 Yes!

What is the expression for the sequence in the table below? Position 1 2 3 4 Value 9 14 ? The value goes up by 5, so its 5x. But the next term is 1 less, so the expression is 5x – 1!

Who uses patterns and functions?

Functions are used in business to show costs Functions are used in business to show costs. For example: Snoopytronics Inc. needs to hire a company to monitor employee computer activity. The cost is $4 for each computer.

Hmmm, I wonder what the rule would be… If we make a table of input and output values of cost per computer, it would look like this: Input (x) output(y) 1 2 3 4 Hmmm, I wonder what the rule would be… 4 8 12 16

The rule can also be written as an algebraic equation, such as y = 4x, where y is a function of x.

We can make a graph of this “rule” by using the x and y as coordinates. y = 4x Input (x) output(y) 1 2 3 4 16 12 8 Input (x) output(y) 1 2 3 4 16 12 8 4

Let c = the number of CD’s burned and Suppose Snoop’s band recorded a CD. They want to make copies and sell them. They found a recording company that charges $250 for making a master CD and designing the art for the cover. In addition, they charge $3 to burn each CD. The total cost of making CD’s depends on the number of CD’s burned. Let c = the number of CD’s burned and t = the total cost. What would be the rule?

Yes! The rule would be… t = 3c + 250

You can also write a rule from a graph using the coordinates! Input (x) output(y) -1 1 2 3 5 Hint: make a table! Y = 2x-1

Now, let’s try some examples!

What is the rule for the table below? Correct! 5x

Which equation fits the table? y = x + 15 y = 10x + 5 y = 3x + 12 y = 15x

What’s the rule for this one? Correct! Divide by 2, or y = 1/2 x

What is the rule for the sequence in the table below? Position (x) 1 2 3 4 Value (y) 13 21 29 37 Correct! 8x + 5

What would be the value for y if x = 4? input(x) output(y)   1 2 3 7 9 11 X = 15 13

See if you can guess the rule. Correct! y = 4x + 1 input(x) output(y) ?   1 2 3 1 5 9 13

I’ll give you a hint ~ make a table! Can you guess the rule? I’ll give you a hint ~ make a table! input(x) output(y) 1  0 2  2 3  4 4  ?

I need to workout. Snoopy is thinking of joining a health club. It charges a one-time membership fee of $30 plus a monthly fee of $12. Write a rule for the total charge (y) for the number of months (x) Snoopy goes. Correct! y = 12x + 30

Now, it’s your turn to create a real-world problem with a rule, a matching function table, and a coordinate graph of the rule. Write it in your journal.