Warmups! Write down & Plot the points on the graph

Transcription

1 Warmups! Write down & Plot the points on the graph 1) 3,5 2) 2, 4 3) 3,6 4) 0, functions All you need to know about function and relations. 1

2 Goal: Understand relations and functions Review what we did last class Talk about functions, function rules and variables. Talk about relations vs. functions The function and what the devil is it??? Input/output models. Vertical line test! Using tables to figure out outputs. Functions and Graphs Last Class: Plotting points on a coordinate plane. We covered what an ordered pair was. A general form of an ordered pair is,. Ordered pairs are on a graph, and tell us where to put the point. Now we need to focus on how lines and points get onto a coordinate plane from an equation. Lets talk about the machine that puts points on a graph. 2

3 Function: The machine that changes x. The game analogy: Does anyone play a game on their phone or entertainment system that deals with money or currency or something you buy with? Gain money: Lose money: A function is the same thing! It adds or subtracts or multiplies or divides the x and changes it! Games: Clash Royale Suppose there was a variable called gems. Stuff can happen to your gems, for instance, you may spend it on an upgrade for one of your units. This is like a function: 100 What does this upgrade do? Suppose you open a chest to find more gems! Lets write this in function notation: 5 What does the chest function do? 3

4 Italian sausage maker Who here know how sausage is made? You put ground pork into a vessel, grind it up, then it pushes through a tube into a pig intestine to create sausage. The same thing happens with x!! What happens in the machine: that is all the,,, stuff. Lets see what happens Yum Function notation just tells you what happens to x to change it. Now lets move back to math. Think of the same thing, but instead of gems, we use x. means a function of x, or something happens to x to change it. 2. This means that x will be multiplied by 2 to be changed! is called the input, and is called the output. You put x into the function, and y is your output. 4

5 Relations, inputs and output. When you plug in a number into a Relation, another number pops out! How do we know what goes in and what comes out? ORDERED PAIRS!!, Example: 9, 33. What is the input? What is the output? What is the difference between a relation and a function? Lets answer that on the next slide. Relations vs Function Relations are simply an input and an output. Anything works! 2,4, 54,100, 2,1 A function is a special type of relation: it produces a unique, one of a kind, output. You put in a number, its spews out only one unique number. Function Rule: for every input, there must be 1, and only 1, output!! How do we see this? How do we know?? 5

6 Examples!! There are 5 ways. Ordered Pairs: 3,5 Mapping diagrams: Tables: Graphs: Equations (we will get to this later with a table) Ordered pairs: To determine if the relation is a function, we check to see if all x values produce 1 and only one y value, (AKA: are their two ordered pairs with the same x value and different y values?) Example: Is the relation a function? 2,3, 3,8, 2,9, 4,5, 1,4 With this example, we can see that -2 has two answers!!! 2 3 & 9. Therefore: Note: the y s can be the same, but the x s can t! Another example: is the relation a function? 4,3, 3,2, 5,3, 7,9, 10,11, 9, 1 What do you think about this? 6

7 Mapping diagrams Mapping diagrams just say where does the x go. If the x goes to two y values, the relation is not a function. The arrows point to where the x goes. The example below is a mapping diagram. What do you notice about the number 3 and what does this mean?? Input output tables This is similar to a T-Chart. If the x repeats and gives two different y values, then the relation is not a function. Otherwise, it s ok! Remember, the y s are allowed to repeat, but x s can t. What do you think about a & b? Which one is a function and which one is not? 7

8 Graphs A graph can be represented by either dots, or a line. How do we figure out if a relation is a function by looking at a graph? We use: The Vertical Line Test!! (VLT) The rule is, if we can draw a vertical line through the graph and it cross the graph more than once, then it isn t a function!!! Examples!! Function or not??? On your own, decide if the relation is a function! 8

9 Domain and Range (not on test) Domain of a function: These are the set of numbers that you input to a function! These are your x values! Range of a function: These are all of your outputs of the function. These are all your y values! Examples: What is the domain and range of the function? 3,4, 4,5, 5,6, 6,0, 7,5 Domain: Range: Graphically: We look at the max and the min. (Not on test) The domain and the range from a graph are decided by your maximum point and minimum point for both x and y. Example: Find the domain and range of the line. Domain: Minimum x value is? Max x values is? X is in between those, and equal to! Range: Min y value? Max y value? Y is in between those, and equal to! 9

10 Domain and range of a graph with dots. With dots, we list out the values of x for the domain, and the values of y for the range. Example: Find the domain and range. Independent variable and dependent variable Independent variable (x): this is the input value. Dependent variable (y): this is the output. So, what are all of the names of the input and output? Lets list them!! Volunteers?? Input value names vs. output value names 10

11 Using a table to find your outputs Suppose we have a function: 2. We may want to find out some of our outputs by plugging in some inputs. We say, if x = 1, then what does our output (y) equal? input: x Relation (work, plug it in!) Output: y Input / Output Tables Lets work on some function with tables. Fill in the table for 5 solutions to the function. 3 Input: x Relation (work, plug it in!) Output: y = f(x)

12 Last time! Lets work on some function with tables. Fill in the table for 5 solutions to the function. 4 Input: x Relation (work, plug it in!) Output: y=f(x) p.108 #1-16 (no need to do domain and range) 12