Functions
Lesson 6
We have a warm up on patterns, we are trying to find the next three numbers that will come in the pattern and identify what the pattern is. We are going to “S” our problem.
“Robert did 25 push ups on Monday. For each of the next 5 days, he did 4 more push ups than the previous day. Using this pattern, how many push ups did Robert complete on Saturday?”
Our first step in “S” studying the problem is to underline the question. Our question is “Using this pattern, how many push-ups did Robert complete on Saturday?” The second step in “S” is answering the question “What is this problem asking me to find?” This problem is asking me to find the number of push up’s completed on Saturday. We will complete this SOLVE problem at the end of the lesson.
We are going to model how a function works for your students using the function machine. Discuss the machine with your students…if they go to an example like a car assembly line every single part of the assembly line completes a function or completes a particular part of putting the car together. One function is to put the wheels on each car, come up and that person puts the wheels on; another car comes up and that person puts the wheels on. Everyday every car the function does the same thing, put the wheel on. Function machines work the exact same way. The first function that we are going to look at is to add 2. If we are adding 2, we can represent this with 2 yellows. We can input numbers into our function machine. When the number comes into the function machine you add 2 and then the entire summer comes out. One example would be to input a 1, if you input 1. 1 comes down into the function machine, you add 2 and you output a 3. When you input a 1 it came down, you add 2, and you output a 3. You can have an input of 2. The function machine goes back to add 2. When you change your input to 2, which will be represented by 2 yellows, the 2 yellows come down into the function machine, you add 2 to those 2 yellows and the output becomes a positive 4. The function machine will work again, and its function is add 2. If you input a 5 into the function machine, you will represent that 5 with 5 yellows. The 5 yellow’s will come down into the function machine, you will add 2 to the 5 and you will output 7 yellow’s. So our output will be 7. You can input “0” into the function machine. A “0” is input, you still add 2, and your output is going to be 2. You can input a negative into a function machine. If you input a negative one what happens you have 1 yellow and 1 red, zero pair, so it cancels and your new output is one. The key to a function machine is that the same function occurs every single time. If I input 1 again, 1 comes down into the function machine and I output 3. If I input I again, one comes down into the function machine and my output is 3. Every time you input 1 the output is 3 b/c the same function happens each time. My first function was add 2.
We are going to model the function subtract 1. When we work with integers our rule tells us that when we subtract we can also add the opposite. We can change subtract 1 to adding the opposite of 1 which is negative 1. To represent adding a negative 1. To represent adding a negative 1 we will put 1 red inside our function machine. We are now going to input numbers so that we can get an output. If we were to input 1. The 1 would go down into the function machine. 1 and negative 1 are a zero pair. So they will cancel and our output will be zero. Our function is add negative 1, so we go back to having 1 red chip. If we input a 2, the 2 yellows come down into the function machine, we can’t have an answer with 2 different colors so we remove the zero pair and we output 1 yellow. Our function of subtract 1 still adding negative 1. If we input negative 1 our input would be 1 red and it will come down into the function machine. Right now we have all one color so our output will be negative 2. If we input 1 again, which is what we originally input, we still create 1 zero pair which leave us with an output of zero. A function does the same thing every single time. If our function is 2 times the input minus 2 and we have an input of negative 5. We can use the integer chips to model what our output would be. The first part of the function says 2 times the input which means 2 groups of negative 5. The second part of the function says minus 2 we need to take away a positive 2 or 2 yellows. We do not have any yellows to take away. If we want to take away something that is not there, we have to create the possibility with zero pairs. Adding 1 yellow and 1 red will still give us the value of negative 10, but it gives the opportunity to take away our minus 2, when we take away those 2 you are left with the answer of 12 reds or negative 12. If our input is negative 5, our outputs negative 12.
We are going to complete the table for the function, “Two times the input minus 2”. If I input this negative 4 we are going to multiply the input times 2. 2 times negative 4 and then we are going to subtract 2. 2 times negative 4 is equal to negative 8 and negative 8 minus 2. You are going to add the opposite so that negative 8 plus negative 2 equals a negative 10. We have an input of negative 2 we have 2 times negative 2 minus 2. Multiply first in order of operation so 2 times negative 2 is negative 4, so when you subtract 2 you ad the opposite. A negative 4 plus negative 2 is a negative 6. When we have 2 times negative 1 subtract 2…2 times negative 1 is negative 2 and you are misusing 2 you are going to change the subtraction to adding the opposite. A negative 2 plus negative 2 is negative 4. When we have zero as our input, we have 2 times zero which is zero. And zero minus 2 equals and output of negative 2. We input 1, we have 2 times 1 minus 2, which equals 2 minus 2 or and out put of zero. When we have 2 times and input of 2 minus 2, 2 times 2 is 4 and 4 minus 2 equals 2. We have 2 times the input of3 minus 2 and 2 times 3 equals 6 and 6 minus 2 equals 4. If we have 2 times 5 minus 2; 2 times 5 equals 10 and 10 minus 2 equals 8. We have created out input output chart and now we are going to graph the points on our function. Our first point is negative 5 negative 12. When we graph a point we always start at zero, zero; the negative 5 tells us we are going to move to the left 5 units; 1, 2, 3, 4, 5. The negative 12 tells us we are going to move down 12 units. Our first point negative 5, negative 12. To graph our next point negative 4, negative 10, we start at zero, zero and move to the left 1, 2, 3, 4 units and down 10. Our next point is negative 2, negative 6. Start at zero, zero and we move to the left 2 units, and a negative 6 tells us to move down 6 units. Negative 1, negative 4 start at zero, zero, move to the left 1 and down 4 units. Zero, negative 2; start at zero, zero and we move down 2 units. One, zero we move to the right 1 and we do not move up or down at all. Two, two we move to the right 2 units and up 2 units. Three, Four move to the right 1, 2, 3 units and up 1, 2, 3, 4 units. Five, Eight start at zero, zero go to the right 5 units and up 8. If you notice all of our points are in a line. We are going to see that all our points can be connected with a line. Our input X-value and our output for our Y-value can be used to write the relationship which represents the line. Our original function was “two times the input minus two”. Our output is equal to 2 times the inputs minus 2. So Y equals 2 times X minus 2, which can also be written as just 2X minus 2.
To determine whether a set of points is a function or not we have to keep in mind what we originally said about functions. The same thing happens to the numbers every single time. What you input the same operation happens so the output will always be the same. If you input negative 3, the output will always have to be negative 7. If you input negative 2, the output always hast to be negative 6. Because of all our inputs are unique or not the same, this set of points is a function. We can identify the domain and the range of the function. The domain is the same as the X-coordinate, another word for domain is input, another word for domain is independent variable. The domain in this set of coordinates is negative 3, negative 2, negative 1, zero, 1, 2, and 3. The range of a function is the set of all the Y-values. The range could also be considered the output or the dependant variable. The range for this function is negative 7, negative 6, negative 5, negative 4, negative 3, negative 2, and negative 1. The range is the dependant variable, it depends on what you input for X. If you input 3 depending on what your function is…your Y depends on the 3 that you input. The range is the dependent variable and the domain or input is the independent variable. If we are going to complete a table and graph the rule Y equals 2 X minus 2. We have the input of negative 1, our input is the same as X, so if our rule is 2 times X, we will have 2 times negative 1 minus 2. 2 times negative 1 is negative 2 and negative 2 minus 2 is negative 4. The ordered pair is X comma Y. So our X is negative 1 and our Y is negative 4. If we have an input of zero, we have 2 times zero minus 2. 2 times zero is zero and zero minus 2 is negative 2. So our ordered pair is and input is zero and an output of 2. If we have an input of 1 our rules says 2 times our input of 1 minus2. 2 times 1 is 2 and 2 minus 2 is zero. So we have and ordered pair of 1 comma zero. To graph ordered pairs we will be gin at zero, zero. Our first ordered pair is negative 1 negative 4. So we move to the left 1 unit and down 1, 2, 3, and 4. Our second ordered pair is zero negative 2. So we do not have to the left or right at all and we move down 2 units. Our third ordered pair is 1 comma zero. So we move to the right 1 unit but we do not move up or down at all.
When we write a function so far we have used the variables X and Y. We said that X was our independent variable and we could plug different values in for X; and that would give you your Y value, or your dependent variable. The function Y equals 2 X plus 3 can also be written as F of X equals 2 X plus 3. The function of X equals 2 X plus 3. This gives us another way to write a function and it also tells us another way to evaluate functions. If we wanted to find out what the Y value is if X equals 1; we can always say F of 1 equals 2 X plus 3. This tells us that instead of X we are going to plug in 1 for the X. F of 1 must be equal to 2 times 1 plus 3. F of 1 will be equal to 2 plus 3 or 5. A function such as Y equals 3 X can also be written as the function of X equals 3 X or of X equals 3 X. If we are gong to graph the function Y equals 3X plus 4; we have our inputs for X and we need to determine our Y. If you input a negative 3, you 3 times negative 3, which is negative 9 plus 4, which is negative 5. 2 times 3 is negative 6 plus 4 is negative 2. 3 times an input of negative 1 is negative 3 plus 4 is 1. 3 times zero is zero plus 4 is 4. 3 times 1 is 3 plus 4 is 7. 3 times 2 is 6 plus 4 is 10. 3 times 3 is 9 plus 4 is 13.
If we are going to graph these point we will begin at zero, zero. Our first point negative 3 negative 5 will move to the left 3 units and down 5 units. Our next point negative 2 negative 2; we will move to the left 2 units and down 2. Negative 1, 1 lets move to the left 1 and up 1. Zero, 4 we will not move left or right and we will move up 4. And 1, 7 will be above the top of our graph. We can see that these points when are when are graphed form a line. This is a linear function. I am going to give you the values of Y 18, 8, 2, 0, 2, 8, and 18. We are going to graph the points that will fit on our coordinate plane. Our coordinate plane is from negative 12 to positive 12. So we will not be alb to graph 3, 18. Negative 2, 8 we will be able to graph. If we start at zero, zero. Negative 2 tells us to go to the left and the 8 tells us to go up 8 units, 5, 6, 7, 8. Negative 1, 2 tells us to go to the left1 units and up 2 units. Zero, zero is our origin and we will put a point at zero, zero. 1, 2 mean we go to the right 1 and up 2. And 2, 8 means we go to the right 2 units and up 8 units. If we were to try to use a ruler it would be impossible to connect all 5 points with a straight line. Does this relation or set of ordered pairs form a straight line? Then it’s non-linear. This relation is non-linear. Is this relation a function? From what we know about functions we can see that none of the inputs have different outputs. We don’t have another negative 3 that goes to a different place. We have 3 that goes to 18 and we have 3 that goes to 18, but our problem would be if we had negative 3 going to 18 here and 3 going to 8 in another spot, so because all of our inputs go to a unique output this is a function.
If we only have a picture of a graph or a relation we can still determine if it is a function or not. We can use what is called the Vertical Line Test. The Vertical Line Test says that if you move a vertical line from left to right, if the vertical line touches more than one point at any one time it is not a function. Looking at problem one you can use a pen pencil as your vertical line and as I move it from left to right, the pen intersects the relations only one time so number 1 is a function. It is non-linear because it does not form a line but it is a function. Number 2, if I move my pen across the function it only intersects in one point at any time, so number 2 also passes the Vertical Line Test and it is a function. And because it forms a line it is a linear function. Looking at our 2 graphs again we know that both of them pass the Vertical Line Test. So number one is a function however it is non-linear. Number 2 is a function, but it is linear because it forms a line. To use our Vertical Line test for number 3, if we move our pen or pencil from left to right ; you will see the pen is intersected by the function in 2 places, when there is more than one intersection point of the vertical line we know that this is not a function. Number 3 is not a function. Number 3 also does not form a line so number 3 is non-linear. Number 4 we will use our Vertical Line Test and run our pen or pencil from left to right seeing that it does not ever touch the pen in more than one place. Number 4 is a function however it is not linear because it is not a line, so number 4 is also non-linear.
This problem is asking me to fine the number of pushups completed on Saturday. In “O” organize the facts, we are going to identify each fact and then decide whether it is necessary or unnecessary.
Robert did 25 pushups on Monday, this is a fact; for each of the next 5 days, he did 4 more pushups than the previous day. We now have 2 facts: Robert did 25 pushups on Monday, which is necessary and our second fact is that it increases by 4 pushups everyday for 5 days.
In “L” we are going to line up a plan. We are going to write in words what your plan of action will be. There are two options modeled in the teacher guide for this problem. I’m going to model using the operation of addition. “We are going to add the increase in pushups to the original number of pushup for the number of days.”
In “V”, verify your plan with action. First we will estimate. We know that Robert is going to be doing more than 25 pushups. So our estimate could be: more than 25. In order to model our “L” we are going to add the increase in pushups to the original number of push ups. The increase each day was 4 push up. We originally started with 25. If we increase for 1 day we are going to add 4 which give us 29. Four our second day we are going to add 4 which give us 33. Our third day we are still increasing by 4, so we are at 37 pushups, our fourth day we are gong to add 4, so we will be at 41 pushups. And our fifth day we will add 4 and we will be at 45 pushups.
In “E” examine the results. You will first go back to your questions. Did you answer what you were asked to find? Our questions was “using a pattern, how many pushups did Robert complete on Saturday?” He completed 45 pushups. So our answer is what we were asked to find. Are you answer reasonable? We will check our estimate, our estimate was more than 25. So our answer was reasonable. And is your answer accurate. Your student will rework the problem in another place or check their answer with a calculator. Our answer is accurate. We will write our answer in a complete sentence. Robert completed 45 pushups on Saturday.
To close your lesson you will review the essential questions.
One how does a function work? Given a rule the input value will determine the output value. You substitute the input value into the function and solve to determine the output value.
Number 2, what letter is used to represent the input? X is used to represent the input.
Number 3, what letter is used to represent the output? Y is used to represent the output.