Pre-Algebra Notes Unit 8: Graphs and Functions

Transcription

1 Pre-Algebra Notes Unit 8: Graphs and Functions The Coordinate Plane A coordinate plane is formed b the intersection of a horizontal number line called the -ais and a vertical number line called the -ais. The -ais and -ais meet or intersect at a point called the origin. The coordinate plane is divided into si parts: the -ais, the -ais, Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. (Refer to the diagram on the net page.) Hint: a wa to remember the order of the quadrants is to think of writing a C (for coordinate plane) around the origin. To create the C ou start in quadrant I and move counterclockwise (and so does the numbering of the quadrants). -ais Quadrant II Quadrant I Quadrant III -ais Quadrant IV The coordinate plane consists of infinitel man points called ordered pairs. Each ordered pair is written in the form of (, ). The first coordinate of the ordered pair corresponds to a value on the -ais and the second number of the ordered pair corresponds to a value on the -ais. Our movements in the coordinate plane are similar to movements on the number line. As ou move from left to right on the -ais, the numbers are increasing in value. The numbers are increasing in value on the -ais as ou go up. To find the coordinates of point A in Quadrant I, start from the origin and move units to the right, and up 3 units. Point A in Quadrant I has coordinates (, 3). To find the coordinates of point B in Quadrant II, start from the origin and move 3 units to the left, and up units. Point B in Quadrant II has coordinates ( 3, ). To find the coordinates of the point C in Quadrant III, start from the origin and move units to the left, and down units. Point C in Quadrant III has coordinates (, ). To find the coordinates of the point D in Quadrant IV, start from the origin and move units to the right, and down units. Point D in Quadrant IV has coordinates (, ) origin (0, 0) B A C D -6 McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 1 of 1 Revised CCSS

2 Here are a few phrases that teachers have used with students to help them remember how to determine the coordinates of a point. Tai before ou take off implies moving right or left before ou move up or down. Same with Run before ou jump. Notice that there are two other points on the above graph, one point on the -ais and the other on the -ais. For the point on the -ais, ou 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 -ais, ou do not move left or right, but ou do move up units on the -ais. This point has coordinates of (0, ). Points on the -ais will have coordinates of (, 0) and points on the -ais will have coordinates of (0, ). The first number in an ordered pair tells ou to move left or right along the -ais. The second number in the ordered pair tells ou to move up or down along the - ais. Relations and Functions Sllabus Objective.3: The student will represent relations and functions using graphs, tables and mapping diagrams. CCSS 8.F.1: Understand that a function is a rule that assigns to each input eactl one output. Understand the graph of a function is the set of ordered pairs, consisting of an input and the corresponding output. (Function notation is not required.) Students that have read a menu have eperienced working with ordered pairs. Menus are tpicall written with a food item on the left side of the menu, the cost of the item on the other side as shown: Hamburger.$3.0 Pizza Sandwich...00 Menus could have just as well been written horizontall: Hamburger, $3.0, Pizza,.00, Sandwich,.00 But that format (notation) is not as eas to read and could cause confusion. Someone might look at that and think ou could bu a $.00 sandwich. To clarif that so no one gets confused, I might group the food item and its cost b putting parentheses around them: (Hamburger, $3.0), (Pizza,.00), (Sandwich,.00) Those groupings would be called ordered pairs because there are two items. Ordered because food is listed first, cost is second. B definition, a relation is an set of ordered pairs. McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page of 1 Revised CCSS

3 Another eample of a set of ordered pairs could be described when buing cold drinks. If one cold drink cost $0.0, two drinks would be $1.00, and three drinks would be $1.0. I could write those as ordered pairs: (1,.0), (, 1.00), (3, 1.0), and so on. From this ou would epect the cost to increase b $0.0 for each additional drink. What do ou think might happen if one student went to the store and bought drinks for $.00 and his friend who was right behind him at the counter bought drinks and onl paid $1.7? M guess is the first gu would feel cheated, that it was not right, that this was not working, or this was not functioning. The first gu would epect anone buing four drinks would pa $.00, just like he did. Let s look at the ordered pairs that caused this problem. (1,.0), (, 1.00), (3, 1.0), (,.00), (, 1.7) The last two ordered pairs highlight the malfunction, one person buing drinks for $.00, the net person buing drinks for a $1.7. For this to be fair or functioning correctl, we would epect that anone buing four drinks would be charged $.00. Or more generall, we would epect ever person who bought the same number of drinks to be charged the same price. When that occurs, we d think this is functioning correctl. 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 elements, those ordered pairs would not be classified as a function. If I asked students how much 10 cold drinks would cost, man might realize the cost would be $.00. If I asked them how the got that answer, eventuall, with prodding, someone might tell me the multiplied the number of cold drinks b $0.0. That shortcut can be described b a rule: cost = $0.0 number of cold drinks c =.0n or the wa ou see it written in our math book 1 =.0 or = That rule generates more ordered pairs. So if I wanted to know the price of 0 cold drinks, I would substitute 0 for. The result would be $ Written as an ordered pair, I would have ( 0,10 ). Note: You can also use a couple eamples to eplain wh ou can have the situation (1, $1), (, $1), (3, $30), (, $30). Sometimes there can be the same value for different McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 3 of 1 Revised CCSS

4 values. I tell them it is like bu one, get one free one and two shirts still cost $1, but the 3 rd and th shirts will cost $30 and this still follows the function rule. Also eplain the situation (1, $10), (, $10), (3, $10), (, $10) where all the values are the same but the value is different b using the all ou can eat eample. At Cici s pizza, ou can bu the all ou can eat special for $10. If ou eat one piece of pizza, it is $10. If ou eat 3 pieces of pizza, it is still $10. If ou eat 100 pieces of pizza, it is still $10. The function rule remains in place! Let s look at another rule. Eample: = 3+ If we plug in, we will get 1 out, represented b the ordered pair (, 1). If we plug 0 in, we get out, represented b (0, ). There is an infinite number of numbers I can plug in. A relation is an 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. Sometimes we put restrictions on the numbers we plug into a rule (the domain). Those restrictions ma be placed on the relation so it fits real world situations. For eample, let s use our cold drink problem. If each drink costs $0.0, it would not make sense to find the cost of drinks. You can t bu negative two drinks, so we would put a restriction on the domain. The onl numbers we could plug in are whole numbers: 0, 1,, 3 The restriction on the domain also affects the range. If ou can onl 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 ever member of the domain, there is one and onl one member of the range. That means if I give ou a rule, like = 3, when I plug in =, I get out. This is represented b the ordered pair (, ). Now if I plug in again, I have to get out. That makes sense it s epected so just like the rule for buing cold drinks, this is working, this is functioning as epected, it s a function. Now, ou are thinking, big deal, isn t that what we would epect? Looking at another rule might give us a clue, + =. Solving that for, we get = = ± McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page of 1 Revised CCSS

5 Now if we plug in a number like 3, we get two answers: (3, ) and (3, ). You can see there is not one and onl one member in the range for each member in the domain. Therefore, this rule describes a relation that is not a function. We can look at the graphs of relations that are nothing more than a bunch (set) of ordered pairs (points) and determine if it s a function. To determine if a graph describes a function, ou use what we call the vertical line test. That is, ou tr to draw a vertical line through the graph so it intersects the graph in more than one point. If ou can do that, then those two ordered pairs have the same first element, but a different second element. Therefore, the graph would not describe a function. If there does not eist an vertical line which crosses the graph of the relation in more than one place, then the relation is a function. This is called the VERTICAL LINE TEST. What we tr to do is draw a vertical line so it intersects the graph in more than one place. If we can t, then we have a graph of a function. Let s tr a few problems. Label the following as relations or functions. A B C D E F Onl A and E are functions. In addition to looking at a graph to represent a relation, ou can use a mapping diagram. McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page of 1 Revised CCSS

6 Eample: Represent the relation ( 1, 1 ), ( 3, 0 ), ( 3, 1 ), (, 3 ), ( 1, ) using a mapping diagram. List the inputs and the outputs in order. Draw arrows from the inputs to their outputs. Input Output Is this relation a function? No, because the input 3 is paired with two outputs, 0 and 1. You could also verif this b graphing the ordered pairs and appling the vertical line test. Graphing Linear Equations Sllabus Objectives: (.) The student will find solutions of linear equations in two variables. A linear equation is an equation whose solutions fall on a line on the coordinate plane. All solutions of the linear equation fall on the line, and all the points on the line are solutions of the equation. In order to plot the graph of a linear equation, we solve the equation for in terms of. Then we assign values for and find the value of that corresponds to that. Each and, called an ordered pair (, ), represents the coordinate of a point on the graph. Eample: Graph 3+ =. Solving for, I subtract 3 from both sides. = 3 or = 3+ When I assign values for, I get these values: Rewriting as ordered pairs, I get (0, ), (, ), ( 1, ) which I plot on the graph. When I connect those three points, I get a straight line called a LINEAR equation. McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 6 of 1 Revised CCSS

7 I could have chosen an values for and found the corresponding. However, it is easier to choose a convenient number, like zero, one, or two. Choosing a number like 100 would make m graph a lot larger; or I could have chosen a fraction, but that can be mess. Slope Sllabus Objective.: The student will find the slope of a line from a graph, equation, or set of ordered pairs using the slope formula. The idea of slope is used quite often in our lives. However, outside of school, it goes b different names. People involved in home construction might talk about the pitch of a roof. If ou were riding in our car, ou might have seen a sign on the road indicating a grade of 6% up or down a hill. Both of those cases refer to what we call slope in mathematics. Kids use slope on a regular basis without realizing it. Let s look at our drink eample again. A student bus a cold drink for $0.0. If two cold drinks were purchased, the student would have to pa $1.00. I could describe that mathematicall b using ordered pairs: (1, $0.0), (, $1.00), (3, $1.0), and so on. The first element in the ordered pair represents the number of cold drinks; the second number represents the cost of those drinks. Eas enough, don t ou think? Now if I asked the student how much more was charged for each additional cold drink, hopefull the student would answer $0.0. So the difference in cost from one cold drink to adding another is $0.0. The cost would change b $0.0 for each additional cold drink. The change in price for each additional cold drink is $0.0. Another wa to sa that is the rate of change is $.0. We call the rate of change slope. In math, the rate of change is called the slope and is often described b the ratio rise run. The rise represents the change (difference) in the vertical values (the s); the run represents the change in the horizontal values, (the s). Mathematicall, we write 1 m = 1 Let s look at an two of those ordered pairs from buing cold drinks: (1, $0.0) and (3, $1.0). Find the slope. Substituting in the formula, we have: m = = = 0.0 We find the slope is $0.0. The rate of change per drink is $0.0. McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 7 of 1 Revised CCSS

8 Eample: Find the slope of the line that connects the ordered pairs (3, ) and (7, 1). 1 To find the slope, I use m =. 1 Subtract the values and place that result over the difference in the values = The slope is. 7 3 For eample, given a linear function represented b a table of values and a linear function represented b an algebraic epression, determine which function has the greater rate of change. Eample: Find the slope of the line on the graph to the right. Pick two points that are eas to identif. (0, 1) and (, ) 1 Find the slope: m = 1 1 = 0 3 = The slope of the graphed line is CCSS 8.F.: Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions. For eample, given a linear function represented b a table of values and a linear function represented b an algebraic epression, determine which function has the greater rate of change. The following is adapted from the Arizona Academic content Standards, 010. McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 8 of 1 Revised CCSS

9 Eample: Compare the two linear functions listed below and determine which function represents a greater rate of change. Function 1: Function : The function whose input and output are related b = 3+ 7 Solution: Function 1 has a slope (rate of change) of or. Function has a slope of 3. The 1 greater the slope, the greater the rate of change, so Function has the greater rate of change. (Steeper slope indicates greater rate of change.) Eample: Compare the two linear functions listed below and determine which has a negative slope. Function 1: Gift Card Samantha starts with $0 on a gift card for the book store. She spends $3.0 per week to bu a magazine. Let be the amount remaining as a function of the number of weeks, Function : The school bookstore rents graphing calculators for $ per month. It also collects a non-refundable fee of $10.00 for the school ear. Write the rule for the total cost (c) of renting a calculator as a function of the number of months (m). Solution: Function 1 is an eample of a function whose graph has negative slope. Samantha starts with $0 and spends mone each week. The amount of mone left on the gift card decreases each week. The graph has a negative slope of 3., which is the amount the gift card balance decreases with Samantha s weekl magazine purchase. McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 9 of 1 Revised CCSS

10 Function is an eample of a function whose graph has positive slope. Students pa a earl nonrefundable fee for renting the calculator and pa $ for each month the rent the calculator. This function has a positive slope of which is the amount of the monthl rental fee. An equation for Eample could be c = m Intercepts Sllabus Objective.6: The student will find the intercepts of a line from a graph, equation, or set of ordered pairs. You can also graph a line quickl b choosing the points where the line crosses the aes. To find the -intercept (-coordinate of the point where the line crosses the -ais), substitute 0 for in the line s equation and solve for. To find the -intercept (-coordinate of the point where the line crosses the -ais), substitute 0 for in the line s equation and solve for. Eample: Find the intercepts of the graph 3 = 6. Graph. To find the -intercept, we will let = 0 and solve for. To find the -intercept, we will let = 0 and solve for. Answer: The -intercept is and the -intercept is 3. I can rewrite these as the ordered pairs (, 0) and (0, 3 ). 3 = ( ) = 6 3 = 6 = 3 = 6 30 ( ) = 6 = 6 = 3 Graphing these intercepts I can now graph the line McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 10 of 1 Revised CCSS - -

11 Eample: Graph 3 = 8. Solve for in terms of. 3 = 8 3 = 8 (subtract from both sides) 1 8 = (divide both sides b 3) 3 3 Assign values for, and find the corresponding s. Choose numbers that will make the values for eas to work with The ordered pairs (0, 8 ), (8, 0), and (, 1) represent the points on the graph. 3 If we did enough of these problems, we would see a quicker wa of graphing linear equations. First, all linear equations are graphs of lines; therefore, all we need do is graph two points. Second, we would see that the value of when the graph crosses the -ais is alwas zero. Look at the last two eamples. What s the value of when the graph crosses the -ais? Look at the graphs we ve alread plotted. When the graph crosses the -ais, the value of is zero. -10 Let s look at another eample. First we ll plot the graph as we did in the previous two eamples; then we ll look at a short cut. Eample: Graph 3 = 1. Solve for : 3 = 1 = 3+ 1 (subtract 3 from both sides) 3 = 3 (divide both sides b ) Now assign values for and find the corresponding s. Graph each ordered pair. McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 11 of 1 Revised CCSS

12 Now for the shortcut. Slope-Intercept Form of a Linear Equation Sllabus Objective.7: The student will graph an equation b identifing the slope and - intercept of a line. CCSS 8.F.3-1: Interpret the equation = m + b as defining a linear function, whose graph is a straight line. 3 We have 3 = 1. Solving for, we have = 3. We graphed that and found the graph to cross the -ais at (0, 3 ). We can determine the slope b using two of the points: m =. 0 I know the graph will cross the -ais at (0, 3 ) and has a slope of 3. That means I can locate the point the graph crosses the -ais, called the -intercept; from there, go up 3 units and over units to locate another point on the line. We can now draw the line. If I did enough problems like that, I would begin to notice the coefficient of the is the slope and the constant is the -intercept (where the graph crosses the -ais). That leads us to the Slope-Intercept Form of an Equation of a Line: = m + b, where m is the slope and b is the -intercept. McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 1 of 1 Revised CCSS

13 Let s look at another problem, = To graph that, I would find the graph crosses the -ais at (0, 3) and has slope. That means, I would locate the point the graph crosses the -ais, called the -intercept (0, 3), and from there go up spaces and over one (recognizing slope as rise = ). The graph alwas crosses the -ais when = 0. run Eample: Find the -intercept and slope of = + 1. Graph. Without plugging in s and finding s, I can graph this b inspection using the slope-intercept form of a line. The equation = + 1is in the = m + b form. In our case, the -intercept is 1, written (0, 1), and the slope is. This slope can be written as or To graph, I begin b plotting a point at the -intercept (0, 1). Net, the slope is. So from the -intercept, I go down two and to the right one and place a point. (Or I could have gone up two and to the left one.) Connect those points, and I have m line. - Eample: Find the -intercept and the slope of =. Graph. The graph of that line would cross the -ais at the -intercept (0, ) and has slope. To graph that, I would go to the -intercept (0, ) and from there, go up two spaces and to the right one space The question comes down to: would ou rather learn a shortcut for graphing or do it the long wa b solving for and plugging in values for? McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 13 of 1 Revised CCSS

14 Let s tr one more. Eample: Graph =. The graph of that line would cross the -ais at (0, 0) and has slope 1. To graph that, I would go to the -intercept (0, 0) and from there, go up one space and over one CCSS 8.F.3-: Give eamples of functions that are not linear. For eample, the function A= s giving the area of a square as a function of its side length is not linear because its graph 1,1,,,3,9, which are not on a straight line. contains the points ( ) ( ) ( ) Eample: A square tile has a side length of inches. The area would be given b the area b. Determine if our equation, = is a linear equation. =. Let One could start b completing a table: side length 1 3 area ( ) or Now plot the points; draw the line or curve. 10 When we connect the points, the graph is NOT linear McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 1 of 1 Revised CCSS

15 Eample: Determine which of the functions listed below are linear and which are not linear and eplain our reasoning. = (linear ; fits the form = m + b ) = + 3 (nonlinear; because of the term) A = ( ) (linear ; can be rewritten in the form = m + b ) = π r (nonlinear; because of the term) Horizontal and Vertical Lines Don t forget to address graphing horizontal and vertical lines (students can have problems with these). The graph of the equation = bis the horizontal line through (0, b). The graph of the equation = ais the vertical line through (a, 0). Of course, remind students to simpl set up a table of values if the are confused. If the line is horizontal, the slope is zero. If the line is vertical, the slope is undefined. Eample: Graph =. The graph of the equation = is the horizontal line through (0, ). Or a quick table would give us points to plot, and then we could draw the line. 0 m = = = = Eample: Graph = 3. The graph of the equation = 3 is the horizontal line through (0, 3). Or a quick table would give us points to plot and then draw the line ( ) ( ) 0 m = 3 3 = 0 undefined = McDougal Littell, Chapter 1, Sections 8 HS Pre-Algebra Unit 08: Graphs and Functions Page 1 of 1 Revised CCSS