Math 7 Notes - Unit 4 Pattern & Functions

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1 Math 7 Notes - Unit 4 Pattern & Functions Syllabus Objective: (3.2) The student will create tables, charts, and graphs to extend a pattern in order to describe a linear rule, including integer values. Syllabus Objective: (3.10) The student will graphically represent equations and inequalities in one variable with integer solutions. Please note: The benchmark calendars have included objective 3.10 with this unit, but after careful consideration, we have found it to be misplaced. To graph in a single variable it must be done on a number line and that is not the intent of this unit. It would be better placed with the integer unit we did previously. Be sure to teach it to students prior to the semester exam, but know it is not part of this unit and has not previously been taught. Syllabus Objective (3.3) The student will evaluate algebraic expressions for given values of a variable. Please note: This was taught in Unit 1 but can be connected to finding the y (or output) value when substituting in values for x to find ordered pairs. Points on the Coordinate Plane A coordinate plane is formed by the intersection of a horizontal number line (called the x-axis) and a vertical number line (called the y-axis). It consists of infinitely many points called ordered pairs. The intersection of the two axes is called the origin. The coordinate plane is divided into six parts: the x-axis, the y-axis, Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. Quadrant II Quadrant III y-axis y x Quadrant I x-axis Quadrant IV The axes divide the plane into four regions called quadrants which are numbered I, II, III, and IV in the counterclockwise direction. Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 1 of 19

2 An ordered pair is a pair of numbers that can be used to locate a point on a coordinate plane. The two numbers that form the ordered pair are called the coordinates. The origin is the intersection of the x- and y-axes. It is the ordered pair (0, 0). The ordered pairs are listed in alphabetical order; (x, y) (input, output), (horizontal axis, vertical axis), (abscissa, ordinate), (domain, range). The alphabetical order will help you remember these terms. To find the coordinates of point A in Quadrant I, start from the origin and move 2 units to the right, and up 3 units. Point A in Quadrant I has coordinates (2, 3). B y x -2 C D -6 To find the coordinates of point B in Quadrant II, start from the origin and move 3 units to the left, and up 4 units. Point B in Quadrant II has coordinates ( 3, 4). To find the coordinates of the point C in Quadrant III, start from the origin and move 4 units to the left, and down 2 units. Point C in Quadrant III has coordinates ( 4, 2). To find the coordinates of the point D in Quadrant IV, start from the origin and move 2 units to the right, and down 5 units. Point D in Quadrant IV has coordinates (2, 5). Notice that there are two other points on the above graph, one point on the x-axis and the other on the y-axis. For the point on the x-axis, you move 3 units to the right and do not move up or down. This point has coordinates of (3, 0). For the point on the y-axis, you do not move left or right, but you do move up 2 units on the y-axis. This point has coordinates of (0, 2). Points on the x-axis will have coordinates of (x, 0) and points on the y-axis will have coordinates of (0, y). To plot points in the coordinate plane, you must follow the direction given. The first number in an ordered pair tells you to move left or right along the x-axis. The second number in the ordered pair tells you to move up or down along the y-axis. A Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 2 of 19

3 Example: To plot the point (4, 2), we start at the origin and move four units to the right and then move two units up. Plot the following points on the coordinate plane below: (2, 5), ( 1, 3), ( 3, 2) and (4, 3). Label the points as A, B, C, and D and identify which quadrant each point is located in. Ordered pairs can be written from a chart or table of values. Ex. Write the ordered pairs using the table below. x y The ordered pairs are (1, 6), (2, 7), (3, 8), and (4, 9) Ex. Write the ordered pairs using the table. The ordered pairs are (0, 3), (1, 4), (2, 5), (3, 6), and (4, 7) x y Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 3 of 19

4 If the table or chart is not labeled as x and y values, we can still convert them into ordered pairs. Days of week Hours worked In that chart, the first day, 3 hours were worked. The second day, 7 hours were worked, etc. Writing these as ordered pairs : (1, 3), (2, 7), (3, 6), and (4, 10). Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 4 of 19

5 Functions Students who have read a menu have experienced working with ordered pairs. Menus are typically written with a food item on the left side of the menu, the cost of the item on the other side as shown: Hamburger.$3.50 Pizza Sandwich Menus could have just as well been written horizontally: Hamburger, $3.50, Pizza, 2.00, Sandwich, 4.00 But that format (notation) is not as easy to read and could cause confusion. Someone might look at that and think you could buy a $2.00 sandwich. To clarify that so no one gets confused, I might group the food item and its cost by putting parentheses around them: (Hamburger, $3.50), (Pizza, 2.00), (Sandwich, 4.00) Those groupings would be called ordered pairs because there are two items. Ordered because food is listed first, cost is second. By definition, a relation is any set of ordered pairs. Another example of a set of ordered pairs could be described when buying cold drinks. If one cold drink cost $0.50, two drinks would be $1.00, and three drinks would be $1.50. I could write those as ordered pairs: (1,.50), (2, 1.00), (3, 1.50), and so on. From this you would expect the cost to increase by $0.50 for each additional drink. What do you think might happen if one student went to the store and bought 4 drinks for $2.00 and his friend who was right behind him at the counter bought 4 drinks and only paid $1.75? My guess is the first guy would feel cheated, that it was not right, that this was not working, or this was not functioning. The first guy would expect anyone buying four drinks would pay $2.00, just like he did. Let s look at the ordered pairs that caused this problem. (1,.50), (2, 1.00), (3, 1.50), (4, 2.00), (4, 1.75) Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 5 of 19

6 The last two ordered pairs highlight the malfunction, one person buying 4 drinks for $2.00, the next person buying 4 drinks for a $1.75. For this to be fair or functioning correctly, we would expect that anyone buying four drinks would be charged $2.00. Or more generally, we would expect every person who bought the same number of drinks to be charged the same price. When that occurs, we d think this is functioning correctly. So let s define a function. A function is a special relation in which no two different ordered pairs have the same first element. Since the last set of ordered pairs has the same first element, those ordered pairs would not be classified as a function. If I asked students how much 10 cold drinks would cost, many might realize the cost would be $5.00. If I asked them how they got that answer, eventually, with prodding, someone might tell me they multiplied the number of cold drinks by $0.50. That shortcut can be described by a rule: cost = $0.50 number of cold drinks c =.50n or the way you see it written in your math book 1 y =.50 x or y = x 2 That rule generates more ordered pairs. So if I wanted to know the price of 20 cold drinks, I would substitute 20 for x. The result would be $ Written as an ordered pair, I would have ( 20,10 ). Let s look at another rule. Example: y = 3x+ 2 If we plug 4 in, we will get 14 out, represented by the ordered pair (4, 14). If we plug 0 in, we get 2 out, represented by (0, 2). There is an infinite number of numbers I can plug in. A relation is any set of ordered pairs. The set of all first members of the ordered pairs is called the domain of the relation. The set of second numbers is called the range of the relation. Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 6 of 19

7 Sometimes we put restrictions on the numbers we plug into a rule (the domain). Those restrictions may be placed on the relation so it fits real world situations. For example, let s use our cold drink problem. If each drink costs $0.50, it would not make sense to find the cost of 2 drinks. You can t buy a negative two drinks, so we would put a restriction on the domain. The only numbers we could plug in are whole numbers: { 0,1,2,3,... }. The restriction on the domain also affects the range. If you can only use positive whole numbers for the domain, what values are possible for the range? We defined a function as a special relation in which no two ordered pairs have the same first member. What this means is that for every member of the domain, there is one and only one member of the range. That means if I give you a rule, like y = 2x 3, when I plug in x = 4, I get 5 out. This is represented by the ordered pair (4, 5). Now if I plug in 4 again, I have to get 5 out. That makes sense it s expected so just like the rule for buying cold drinks, this is working, this is functioning as expected, it s a function. Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 7 of 19

8 Graphing Linear Functions Syllabus Objective: (3.6) The student will create coordinate plane graphs to represent linear patterns and nonlinear patterns and functional relationships. Note: Nonlinear patterns is new to the Math 7 curriculum this year. In the past it was accelerated only. Holt users can use Extension section on pages Syllabus Objective: (3.7) The student will graph a set of ordered pairs to represent a linear equation. Syllabus Objective: (3.11) The student will identify linear equations and inequalities. Syllabus Objective: (3.12) The student will model linear equations and inequalities. Syllabus Objective: (3.13) The student will solve linear equations and inequalities using concrete and informal methods. The input value (x-coordinate) is the value substituted into the rule; the output value (ycoordinate) is the value that results from the given input. To find the ordered pairs, you substitute any values of x you choose, then using those values, find the corresponding y-values Ex. Find three ordered pairs that can be written from y = 2x+ 3.. Picking any values of x, 0, 1, and 4, substitute those into the rule, y = 2x + 3 and find the corresponding y values. x 2x + 3 y 0 2(0) (1) (4) The ordered pairs are (0, 3), (1, 5), and (4, 11). Ex. Find three ordered pairs that can be written from y = 3x 2. x y The ordered pairs are (0, 2), (2, 4), and (4, 10). Your ordered pairs could be different, depending on the values of x you choose. Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 8 of 19

9 Remember, those ordered pairs can be graphed on the coordinate plane. A linear equation is an equation whose graph is a straight line. An equation of a line can be written in y = mx + b form. Ex. y = 5x+ 3 is a linear equation. Ex. y = 4x+ 1 is a linear equation To graph a linear equation, you pick any values of x and find the corresponding values for y, make the ordered pair, and then graph those points. Since the graph of a linear equation is a line, you need only graph two points. One convenient point to graph is an ordered pair in which the x coordinate is zero. A linear function is a function whose graph is a nonvertical line. Ex. Graph y = 3x+ 4 When x = 0, y = 4 (0, 4). When x = 1, y = 7 (1, 7) Graph those two points; then draw a line through them. Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 9 of 19

10 Ex. Graph y = 3x+ 2 When x = 0, y = 2 (0, 2). When x = 2, y = 4 (2, 4 ) Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 10 of 19

11 What would have happened if I picked other points to graph based on the rule? Would I have the same graph? The answer is yes! Ex. Graph y = 2x When x = 0, y = 0 (0, 0). When x = 2, y = 4 (2, 4) Graphing Nonlinear Patterns Syllabus Objective: (3.6) The student will create coordinate plane graphs to represent linear patterns and nonlinear patterns and functional relationships. Note: Nonlinear patterns is new to the Math 7 curriculum this year. In the past it was accelerated only. Holt users can use Extension section on pages , as a place to start. If you have studied how to play a musical instrument, played basketball or done any activity that requires practice, you know the more you practice, the better you get. You also know this: In the beginning, a little practice even one session makes a big difference. After a year of regular practice, a single practice session doesn t do nearly as much. The relationship between practice and improvement is nonlinear. If you graphed Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 11 of 19

12 the relationship, you would not get a straight line. In fact, that s why the rate of improvement is called a learning curve and not a learning line. Nonlinear relationships are everywhere. When you drop a penny, the distance it falls in the first moment of its descent is much less than the distance it plummets in the moment before it hits the ground. When you take a drink from a water fountain, the stream forms a smooth curve, not a straight line. A skiers path down a ski jump is a nonlinear relationship. For students the basic understanding of linear versus nonlinear patterns begins easily with a look at some graphs. A. B. B. B. C. Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 12 of 19

13 Having worked with linear patterns students should easily visualize graphs that form a straight line and graphs that do not. Recalling that linear patterns form a straight line, it is easy to expand that to nonlinear patterns do not form a straight line. So, graphs B and C above are nonlinear and graph A is linear. If we examine each of the graphs just seen in the form of a table we begin to see another way to identify linear versus nonlinear patterns. Graph A Input (x) Output (y) The difference is constant. The difference is constant. The function represented has a linear relationship. Graph B Input (x) Output (y) Difference = 5 Difference = 3 Difference = 3 Difference = 5 The difference is constant. The difference varies, is not constant. The function represented has a nonlinear relationship. Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 13 of 19

14 Graph C Input (x) Output (y) Difference = The difference is constant. The difference varies, is not constant. The function represented has a nonlinear relationship. Since the standard states students will create coordinate plane graphs to represent nonlinear relationships be sure to have students graph a few examples. For example, 2 y = x + 1. Have students create a list of ordered pairs and graph the points in a coordinate plane. x 2 y = x + 1 y 3 (-3)(-3) (-2)(-2) (-1)(-1) (0)(0) (1)(1) (2)(2) (3)(3) Once students get the points graphed and realize this graph does not form a straight line, try to get them to connect the points with a smooth graph. (Note: this is the beginning exposure to quadratic functions that are shapes called parabolas.) Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 14 of 19

15 Interpreting Graphs Syllabus Objective: (3.8) The student will model relationships using graphs with and without technology. Syllabus Objective: (3.9) The student will describe relationships using graphs with and without technology. Graphs can be used to show relationships or trends. To do that, you generally use the first quadrant, if there are no negatives numbers, and identify the coordinate axes as reference lines. Suppose you want to make a graph that shows the distance you travel from home to the store and back home again with respect to time. To do that, I would make a graph and label one axis as distance from home and the other axis time. Time Now, as I start to leave home for the store, as time increases, so would my distance from home as shown. When I got to the store, I immediately turn around and come home. My time continues to increase, but my distance from home decreases until I get home. Let s take the same trip to the store, this time taking time to shop. Distance Distance Time Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 15 of 19

16 Notice the beginning line on the graph stays the same as the first graph. That is, the longer I travel, the greater the distance I am from home. The horizontal line shows that my time is still increasing as I am shopping, but my distance from home is staying the same. Then as I am heading home and the time is increasing, the distance from my house decreases. Let s look at another graph. Suppose you are going to take a trip, so you fill your gas tank and drive until the tank is almost empty. Then you fill up the tank and drive again until the tank is almost empty. To draw a graph that represents that, we would label our coordinate axes. One axis we could call gas, the other time. In this graph, we are starting with a full tank, as time increases, the amount of gas in the tank decreases as shown. Gas (in gallons) Time Then, when we are almost empty (notice the line does not touch the horizontal axis), we fill the tank back up. That takes a little time as shown. Then as we continue on our trip, time continues to increase while the gas in the tank decreases again. Note: Be sure to give students graphs and have them write a scenario that represents the graph AND give students a scenario and have them draw the graph to represent it. Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 16 of 19

17 Sequences A sequence is an ordered list of numbers (called terms). In an arithmetic sequence, the difference between one term and the next is always the same. This difference is called the common difference. We can add the common difference to each term to get the next term. Example: Find the common difference in the sequence 7, 11, 15, 19, 7, 11, 15, 19 The terms increase by 4, so the common difference is Example: Find the common difference in the sequence 6, 3, 0, 3, 6, 3, 0, 3, The terms decrease by 3, so the common difference is Example: Find the next three terms in the arithmetic sequence 2.5, 4, 5.5, 7, 2.5, 4, 5.5, 7, Each term is 1.5 more than the previous, so to find the next 3 terms 7, 8.5, 10, The next three terms are 8.5, 10, and In a geometric sequence, the ratio of one term to the next is always the same. This ratio is called the common ratio; each term in a geometric sequence is multiplied by the common ratio to get the next term. For the following examples, determine if the sequence is geometric. If so, give the common ratio. Example: 2, 6, 18, 54, 2, 6, 18, 54 The sequence is geometric with a common ratio of Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 17 of 19

18 Example: 1, 1, 1, 1, ,,, The sequence is geometric with a common ratio of ( 1) ( 1) ( 1) Example: 16, 8, 4, 2, 16, 8, 4, 2, The sequence is geometric with a common ratio of As with the arithmetic sequence, we should know how to extend a geometric sequence. Example: Find the next three terms for the geometric sequence 162, 54, 18, 6, The common ratio is 1 ; therefore the next 3 terms , 2,, 3 9 The next three terms in the sequence would be 2 2 2,, Students should also be able to identify the rule given the sequence. For example, we might be given the sequence 2, 4, 6, 8, and be asked to write the rule that describes the sequence. A table is a very valuable tool to help us with this. We will let n represent the number s place in the sequence (n = 1 for the 1 st term, n = 2 for the 2 nd term, and so on). Then we can fill in the output ( y) values. Now the fun part determining the rule by making guesses and checking to see if they work. Let s start the table for our problem. n Rule y Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 18 of 19

19 We now need to determine what operation or rule would help us to input the n value and output the y value. What would input 1 and output 2? Input 2 and output 4? Hopefully, you might discover multiplying by 2. We can now test the rule: n Rule y 1 1(2) 2 2 2(2) 4 3 3(2) 6 4 4(2) 8 Multiplying by 2 works for all our values. We can now state the rule: y = 2 n. One of the values of knowing the rule is that we can now quickly determine any term in the sequence. Example: What is the 51 st term in the sequence 2, 4, 6, 8,? We know that y = 2 n. If n = 51, then y = 2(51) or 102. The 51 st term of the sequence would be 102. Some functions may involve 2 operations. Look at the example below: Example: What is the 35 th term for the sequence 3, 5, 7, 9,? Let s create a table. n Rule y 1 1(2) (2) (2) (2) Looks like the rule that will work is to multiply by 2 and add one. We can state: y = 2n+ 1. The 35 th term would be 2(35) + 1 or 71. Math 7, Unit 04: Patterns and Functions Holt: Chapter 4 Page 19 of 19

Math 7 Notes - Unit 4 Pattern & Functions

Math 7 Notes - Unit 4 Pattern & Functions Syllabus Objective: (.) The student will create tables, charts, and graphs to etend a pattern in order to describe a linear rule, including integer values. Syllabus