Chapter 4.1 Introduction to Relations

Chapter 4.1 Introduction to Relations The example at the top of page 94 describes a boy playing a computer game. In the game he has to get 3 or more shapes of the same color to be adjacent to each other. If he gets 3 adjacent shapes he earns 6 points, if he gets 4 adjacent shapes he earns 8 points, 5 adjacent points earns 10 points and 6 adjacent shapes earns 12 points. We can show the relationship between the number of shapes in the group and the score earned with ordered pairs. (3,6), (4,8), (5,10), (6,12). (3,6) is not the same as (6,3) Why not? There is a relation between the size of the group and the number of points earned. For 3 shapes you earn 6 points, (6,3) would indicate that for six shapes 3 points were earned which is not the case. We have a name for a relationship represented by a set of ordered pairs. It is called a relation and the first number in the ordered pair is called the domain, the second number in the ordered pair is called the range. In our game example the domain is {3,4,5,6} and our range is {6,8,10,12}. We can express the game relation as a rule, as a table, as a mapping diagram and as a graph. The rule would be expressed how? If (x,y) represent an ordered pair relation in the game then a rule for the relation is y = 2x. Your y value is always 2 times your x value. That is the points earned is 2 times the number of shapes you arrange. X is the input value and y is the output value. Mapping diagram is made with two ovals and you match your x value to your y value. 3 4 5 6 Group 6 8 10 12 Points Table Group (x) Points(y) 3 6 4 8 5 10 6 12

Graph - Not all relationships can be represented by a rule.

Chapter 4.2 Introduction to Functions The problem at the top of page 96 indicates that Mike has a garden with a length of 40 ft. He wants to fence it off so that the deer do not get his plants. We need to use the perimeter formula to determine how much fencing we will need. The formula for the perimeter is P=2l + 2w. We know that the length of his garden is 40 ft so we can rewrite the formula as P = 2(40) + 2w which is P= 80+ 2w. We can then substitute in various values for the width of the garden in order to determine the amount of fencing Mike will have to purchase. The rule for your relation is P=80+2w, the ordered pairs are going to be of the form (w,p) The domain of the relation is going to be all the values for w, the range of the relation is going to be the resulting P values. Can the domain be negative numbers? Why not? Each value we input for w gives us a unique perimeter. The relationship between the width and the perimeter is a function. A function is a special kind of relation in which each domain value is paired with exactly one range value. To determine if a set of ordered pairs is a function you need to check if no first element can be paired with more than one second element. It does not matter if some second elements are repeated. You cannot have (2,3) and (2, 6) to have a function, however you can have (2,3) (4,3) If you are trying to determine if a relation is a function using mapping you need to look at the arrows going from the domain to the range. If one element in the domain has more than one arrow going away from it then it is not a function. If you are looking at graph you can use the vertical line test to determine if a relation is a function. When any vertical line intersects a graph at exactly one point, the graph represents a function. When a vertical line intersects a graph at more than one point the graph does not represent a function. A function rule is an equation that describes a function. In our perimeter example it was P=40+2w. Sometimes it is written in function notation. Function notation, f(x) means that f is a function of x. You use f(x) to replace y. Since the domain or values of x are substituted into the equation to determine the range, x is the independent variable and y or the range is the dependent variable. domain input x x Independent variable range output y f(x) Dependent variable We can write our perimeter function in function notation: f(x) = 80 + 2x. If we have to differentiate amongst functions with the same variable different letters are used in function notation. You can have f(x), g(x), h(x), the f, g and h represent different function rules.

Chapter 4.2A Graphs of Functions Remember that a function is mathematical relationship where every member of the domain (x- value) corresponds with exactly one member of the range (y-value). Getting paid by the hour is a good example of a function. If you make $5 per hour for every hour you work you will receive one and only one salary based on the number of hours you worked. When we are representing a function on the coordinate grid the axes must be appropriately labeled and scaled. You need to choose x values or domain values that make sense for the problem at hand. For our salary example choosing a scale that is hourly meaning increments of 1 hour would work. You would not want to use a scale for hours that was set at 20 hour intervals. Our y axis should be set up with in multiples of $5. If you do not work you make $0, if you work 1 hour you make $5, if you work 1.5 hours you would make $7.50, etc When graphing the above relationship if you have the ordered pair (25, 125) you then know that this means the individual worked 25 hours and made $125. It is also important to realize that the graph of per hour wages will only happen in the first quadrant of the coordinate grid. At no point can you have negative earnings. More than one scale and interval may make sense when graphing certain situations. Scale is the range the x axis covers. If we were looking for per day salary the scale may be 1-8 hours, if we were looking for salary per week the scale might be 8-40 hours. Interval represents the distance between each x value. If we were looking for per day salary the interval might be 1 hour, if we were looking for the per week salary the integral might be 8 hours. f(x) represents the y value on the graph, or the value of the function at a given x value.

Chapter 4.3 Write Function Rules The word problem at the top of page 100 in your textbook describes a man going on vacation for two weeks who has to determine how much it will cost him to hire a dog sitter. The pet shop charges $3 per hour plus a basic two week fee of $50. How can he decide what it will cost him to have his dog taken care of? Let c the total cost of care Let h the total number of hours c = 3h + 50 h is the independent variable because the total cost will depend on how many hours he wants the pet shop to spend with his dog. We can rewrite this in function notation. c(h) = 3h + 50. Using this function we can create a function table. A function table contains the independent variable, the function written in function notation and the dependent variable. Given a function table we can determine the function rule that describes the relationship by looking for a pattern. If the y values are greater than the x values you want to test patterns that require addition and multiplication first. If the y values are less than the x values you want to test patterns with subtraction and division first. Patterns may involve more than one operation or other types of operations such as exponents and square roots. Once you think you have found a rule make sure you check it for all values in the table. A rule that holds for one or two values may not hold throughout. A function table is an organized way to show the values of a function for different inputs. It can also help one write ordered pairs to represent the relationship. These ordered pairs in turn can be graphed on the coordinate grid to have a picture of what the function looks like.

Chapter 4.4 Arithmetic Sequences An ordered set of elements that follows a pattern is called a sequence. 2, 4, 6, 8, 10, 12, 14,.. A sequence with a first term and a last term is called a finite sequence. A sequence in which each term will have another term that follows it is an infinite sequence. We can talk about terms in a sequence as follows: a 1, a 2, a 3, a 4, a 5,...a n-1, a n The subscript in a term represents the terms place in the sequence. The 16 th term is a 16. A sequence in which each term is found by adding a non-zero constant to the previous term is called an arithmetic sequence. The constant we are adding to each term is called the common difference and we represent this with the variable d. To determine if we have an arithmetic sequence we have to check and see if there is a common difference between terms. If there is a common difference then we have an arithmetic sequence. 5, 8, 11, 14, 17, is this an arithmetic sequence? If yes, what is the common difference? 1, 2, 4, 8, is this an arithmetic sequence? If yes, what is the common difference? 7, 3, -1, -5, is this arithmetic sequence? If yes, what is the common difference? Each term of an arithmetic sequence can be related to the first term a. 5, 8, 11, 14, 17, the common difference is 3 so we have to add 3 to each term to get the next term. To find the sixth term in the sequence we start with the first term 5 and add 3 we get the second term which is 8, if we take the second term 8 and add 3 we get the third term 11, if we take the 3 rd term 11 and add 3 to it we get the 4 th term 14, if we take the 4 th term 14 and add 3 to it we get 17, if we take the 5 th term 17 and add 3 to it we get the 6 th term 20. To get from the first term to the 6 th term we had to take the first term and add the common difference to it 5 times. So we have to add the common difference to the first term one less time than the term we are trying to find. To find the 6 th term we take the first term 5 and add 5 times the common difference to it. So a 6 = 5 + (6-1)*3, = 5+(5)*3, = 5+15, = 20 The formula would be a n = a 1 +d(n-1) The nth term of an arithmetic sequence is the sum of the first term a 1, and (n-1) multiplied by the common difference. a n = a 1 + d(n-1), a n = the nth term a 1 = the first term n = the number of terms d = the common difference (d 0) By writing an algebraic expression for the nth term in an arithmetic sequence you are writing a function rule for the sequence.

For our example above: The first term is 5 and the common difference is 3 a n = 5 + (n-1)3 a n = 5 + 3n 3 a n = 2 + 3n a n = 2 + 3n is the function rule, a n is the output and n is the input. The position is the input and the value of the term is the output.

4.5A Recursive Formulas for Sequences Remember that the recursive formula in an arithmetic sequence is: a n = a n-1 + d where a n is the nth term, n is the number of terms and d (d 0) is the common difference. Where a n is the value of the term in the sequence that you are trying to find. a n-1 is the value of the term that precedes the term you are trying to find. And d is the common difference. And the recursive formula for a geometric sequence is: a n = a n-1 *r Where a n is the value of the term in the sequence that you are trying to find. a n-1 is the value of the term that precedes the term you are trying to find. And r is the common ratio. To find the nth term for a geometric sequence you compute as follows: a n = a 1 r n-1 a n where is the term you are trying to find a 1 is the first term in the sequence r n-1 is the common ratio raised to a power one less than the term you are trying to find. And to find the nth term of an arithmetic sequence you compute as follows: a n = a 1 + (n 1)d a n where is the term you are trying to find a 1 is the first term in the sequence You can use sequences and recursive formulas to solve word problems. An example is as follows: Julie bikes 12 miles on Monday. Then she bikes 3 miles every day after that. If she continues this pattern, how many miles will she have biked in all at the end of Friday? We have a pattern with a common difference of 3, every day our distance increases by 3 miles so this is an arithmetic sequence. We can use the formula: a n = a 1 + (n 1)d We need to know how many miles she biked on Friday, or the 5 th day she biked.

Day 1 she biked 12 miles so a 1 is 12. She bikes additional miles each day after that, that is the common difference. a 5 = 12 + (5-1)3 a 5 = 12 + (4)3 a 5 = 12 + 12 a 5 = 24 So on Day 5 she biked 24 miles

Chapter 4-5B Features of Functions In this section we will learn to use a graph in order to describe the domain and range of a function and intervals where the function might be increasing or decreasing. If at a store DVDs sell for $15, the domain of the function is the number of DVDs sold and the range represent the money made in the sale. The amount you make depends on the number of DVDs that you sell. The more DVDs you sell the more money you make. What can be said about his function is that the domain is all whole numbers and that the range is 0 and all multiples of 15. You can determine the domain and range for a function from a graph by looking at the order pairs that lie on the graph of the function. The domain of the above graph is all real numbers because any value of x may be input into the function. The range is also the set of all real numbers because each x value will generate exactly the same y value. Ordered pairs for this function will look like (-1, -1), (0, 0), (.5,.5), The domain for the above graph are the values that fall between -5 and 4. Notice that the graph does not have arrows for endpoints. Arrows indicate that the graph extends in either direction indefinitely. The endpoints indicate that the graph has boundaries. So to indicate the domain of the above graph we would say: (x -5 x 4) or all points along the x axis between -5 and +4. The range for the above function are all the points between -4 and positive 3. We would indicate for the range for the function as (y -4 y 3).

This graph, unlike the first has boundary points. It has a maximum value the domain can be and a minimum value the domain can be as well as maximum and minimum values for the range. When we are working with functions to solve word problems we have to be careful to make sure that our graph only indicates situations that in fact could occur. That means that some graphs will have a maximum and minimum value for the domain and range. In our DVD example the domain will have a minimum of 0 CDs sold, you cannot have a negative value for the domain. The correspond range will also have a minimum. If you sell 0 CDs then you make $0. The graph however will not have a maximum value unless there is a limit on the number of CDs you have to sell.

Chapter 4.5 Geometric Sequences A sequence where each term after the first is found by multiplying the previous term by a constant is called a geometric sequence. The constant in a geometric sequence is called the common ratio and it is represented by the letter r. The common ratio may be found by dividing any term by its preceding term. Ball bouncing example: Bounce # 0 1 2 3 4 Height (feet) 16 4 1 ¼? a 2 /a 1 = 4/16 = ¼ a 3 /a 2 = ¼ common ratio = ¼ an arithmetic sequence is linear and is a function; a geometric sequence is not linear but is also a function. Functions do not have to be linear. Each term of a geometric sequence can be related to the first term a1, and the common ratio, r. The sequence may be written as: a 1, a 1 r, a 1 r 2, a 1 r 3, a 1 r 4,, a 1 r n-1 a 1 = first term a 1 r = second term a 1 r 2 = third term a 1 r 3 = fourth term a 1 r 3 = fifth term a 1 r n-1 = nth term The formula for the nth term of a geometric sequence: a n = a 1 r n-1 a n = nth term a 1 = the first term n = the number of terms r = the common ratio (r 0, r 1) You can use a recursive formula to find the nth term, a n, of a geometric sequence by using the preceding term, a n-1, and the common ratio, r. A recursive formula is a rule for calculating a new term of a sequence from the term preceding it. a n, and a n-1 represent consecutive terms r = the common ratio

r = a n /a n-1 a n-1 *r = a n /a n-1 *a n-1 a n-1 *r = a n a n = nth term r = common factor n = the number of the term Sequence: 8, -28, 98, -343, r = -28/8 = -7/2 a n = -7/2 * a n-1