MATH GRADE 8 UNIT LINEAR RELATIONSHIPS FOR EXERCISES Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : MODELING RUNNING SPEEDS 8.EE.. A Runner A 8.EE.. D sec 8.EE.. D. m/sec 8.EE.. Each line goes through (, ). Therefore, the four lines are proportional relationships that can be represented b the general equation = k, where k is the ratio of an point (, ) on the line. As a result, the equations for each line are: Runner A: = Runner B: = 7 Runner C: =. Runner D: = 8.EE.. Here is an eample of a table that summarizes the times at each specific distance. Runner A Runner B Runner C Runner D m sec.8 sec. sec sec m sec. sec. sec 8 sec m sec 8. sec 9. sec sec 8 m 8 sec. sec.8 sec sec Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : MODELING RUNNING SPEEDS 8.EE.. One equation that hits three hoops is =. 8 = 8.EE. 7. D gal/hr 8.EE. 8. A,8 cars per minutes C cars per minute E, cars per minutes 8.EE. 9. Yes, her pa is a proportional relationship. She is paid at a constant rate, and her income is directl proportional to the number of hours she works. If she divides her income b the number of hours she works, she will alwas get the same number, which is her rate of pa ($/hour). 8.EE.. =, where is Erin s pa ($) and is the number of hours she works 8.EE.. D $ 8.EE.. B Line B has a smaller slope. Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : MODELING RUNNING SPEEDS Challenge Problem 8.EE.. Here is an eample of four possible proportional relationships, two that have positive values of k and two that have negative values of k. Line A: = Line B: = Line C: = Line D: = Line C Line D Line B Line A Lines A and B have positive values of k. Both lines go up from left to right. B contrast, lines C and D, for which the value of k is negative, go down from left to right. k is the ratio of an point (, ) on the line. A positive value of k means that and have either both positive or both negative numbers simultaneousl. On the graph, this means that the lines of proportional relationships with positive k-values are limited to quadrants I and III in the coordinate plane. B contrast, the lines of proportional relationships with negative k-values are limited to quadrants II and IV in the coordinate plane. In order for k to be negative, and must have opposite signs (if is negative, then is positive, and vice versa). Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : INVESTIGATING GRAPHS 8.EE.. B Slope = 8.EE.. A 8.EE.. C Slope = 8.EE.. B (, ) 8.EE.. Slope of line A: Slope of line B: 8.EE.. A Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : INVESTIGATING GRAPHS 8.EE. 7. Proportional relationships have the general equation = k, where k is the slope. Given the slopes, here are the equations of the lines: Line a: = Line b: = Line c: = Line d: = Here are the corresponding graphs. Line c: = Line b: = Line a: = 9 8 7 7 8 9 Line d: = Copright Pearson Education, Inc. 7
Grade 8 Unit : Linear Relationships LESSON : INVESTIGATING GRAPHS 8.EE. 8. a. 8 8 Slope b. The graph represents a proportional relationship because it is a line that passes through the origin, (, ). Its slope is. 8.EE. 9. a. b. = Copright Pearson Education, Inc. 8
Grade 8 Unit : Linear Relationships LESSON : INVESTIGATING GRAPHS Challenge Problem 8.EE.. Line B does not represent a proportional relationship because it does not go through the origin, (, ). The general equation = k does not appl to this line. Suppose ou drew both lines using an slope of our choice for line A (for eample, slope = ) and randoml selected a point on this line as point W, such as W(, ). You would find that line B passes through Z(, ) and that the slope of line B is not., as suggested b Marshall s predictions. Instead, the slope of line B is also, the same as line A. Parallel lines have the same slope; therefore, the slope of line B is the same as the slope of line A. The slope is still If ou draw a right triangle on line B using point Z and the point of intersection of line B with the -ais, ou will find: rise run = = Since = and ( ) = : rise run = 8 Line B Line A Z W rise = run = Copright Pearson Education, Inc. 9
Grade 8 Unit : Linear Relationships LESSON : TRANSLATING GRAPHS 8.EE.. C The slope 8.EE.. a. 8.7. b. The new shifted line does not represent a proportional relationship because the ratio is no longer constant. Also, onl lines passing through the origin, (, ), represent proportional relationships. 8.EE.. No, the resulting line does not represent a proportional relationship because it does not contain the point (, ). 8.EE.. Slope = 8.EE.. A The line was translated units up. C The line was translated 8 units to the left. E Both lines have the same slope. 8.EE.. C = Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : TRANSLATING GRAPHS 8.EE. 7. The equation of the new line is =. = = Suppose (, ) are the coordinates of a point on the translated line and (, ) are the coordinates of the corresponding point on the original line. The original equation is =. The equation of the translated line is = m + b. Since a translated line is alwas parallel to the original line, the slope of both lines is the same. m = = + b You need to find the value of b to figure out the equation of the translated line. You know that a translation of units down means that each point on the translated line has the same -value as the corresponding point on the original line, but a -value that is units less. This relationship can be written mathematicall as: = = If ou replace and b their equivalent epressions using and, ou get: + b = Since = : + b = b = Therefore, the equation of the translated line is: = Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : TRANSLATING GRAPHS 8.EE. 8. The equation of the new line is = + 7. 8 = + = + 7 Suppose (, ) are the coordinates of a point on the translated line and (, ) are the coordinates of the corresponding point on the original line. The original equation is = +. The equation of the translated line is = m + b. Since a translated line is alwas parallel to the original line, the slope of both lines is the same. m = = + b You need to find the value of b to figure out the equation of the translated line. You know that a translation of units up means that each point on the translated line has the same -value as the corresponding point on the original line, but a -value that is greater b units. This relationship can be written mathematicall as: = = + If ou replace and b their equivalent epressions using and, ou get: + b = ( + ) + Since = : + b = + 7 b = 7 Therefore, the equation of the translated line is: = + 7 8.EE. 9. Since the line of a proportional relationship must include the origin, ou know that this graph must have been translated up units, since its current -intercept is (, ). You could also sa that the original line was translated 8 units to the right. Both translations are equivalent. Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : TRANSLATING GRAPHS Challenge Problem 8.EE.. Here is one possible set of five lines. All of the lines are parallel. The are all translations of the same line. From = ou can go up or down, left or right, an amount and the resulting equation will be of the form = + b. This equation represents all the possible translations of =. Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : LINEAR EQUATIONS 8.EE.. B Slope =, -intercept = 8.EE.. D = + 8.EE.. B = 8.EE.. A Line a 8.EE.. B The slope is negative (going down to the right). C The -intercept is positive. 8.EE.. The values of the slope and -intercept can be etracted directl from the equation, which is given in slope-intercept form. slope = m = -intercept = b = This means that the line intersects the -ais at (, ). To draw the line ou need to find a second point on the line. You can use the equation to find the coordinates of an other point. For eample, ou can find the -intercept b replacing with in the equation and finding the corresponding value of : (, ). Knowing that the slope is means that from an point on the line, ou can go up unit and to the right units and end at another point on the line. For eample, if ou start at (, ), another point on the line is ( +, + ) = (, ). = + Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : LINEAR EQUATIONS 8.EE. 7. The values of the slope and -intercept can be etracted directl from the equation, which is given in slope-intercept form. slope = m = -intercept = b = This means that the line intersects the -ais at (, ). To draw the line, ou can use the equation to calculate the -intercept:, Or, ou can use the value of the slope starting from the -intercept to determine that another point on the line is units down and unit to the right: ( +, ) = (, ). The latter point is easier to plot because it uses whole number coordinates. = + Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : LINEAR EQUATIONS 8.EE. 8. The values of the slope and -intercept can be etracted directl from the equation, which is given in slope-intercept form. slope = m = -intercept = b = This means that the line intersects the -ais at (, ). To draw the line, ou can use the equation to calculate the -intercept: 8, Or, ou can use the value of the slope starting from the -intercept to determine that another point on the line is units down and units to the right: ( +, ) = (, ). The latter point is easier to plot because it uses whole number coordinates. = Copright Pearson Education, Inc.
Grade 8 Unit : Linear Relationships LESSON : LINEAR EQUATIONS 8.EE. 9. a. 8 b. The value of the slope can be calculated b appling the formula m = 8 m = = ( ) to the two known points, (, ) and (, 8). The -intercept can be etracted directl from the graph as the -coordinate where the line meets the -ais. b = c. The equation of the line in slope-intercept form is: = + 8.EE.. In a proportional relationship, the ratio of over is constant and the equation can be represented b the general formula = k. In this situation the ratio is not constant, as shown b this table. Number of Items Sold () Monthl Income (),,,,,. The graphs of proportional relationships alwas go through the origin, (, ). But the graph of this relationship does not go through the origin. It intersects the -ais at (,,). Copright Pearson Education, Inc. 7
Grade 8 Unit : Linear Relationships LESSON : LINEAR EQUATIONS Challenge Problem 8.EE. 8.EE.. The relationship is linear but not proportional. This means that its general equation in slope-intercept form is = m + b, where m is the slope and b is the -intercept. The description states that the line intersects the -ais at (, ). This means that b =. The equation so far is = m +. The description states that the line of the non-proportional relationship is parallel to the line of a proportional relationship that intersects (, ). Parallel lines have the same slope. Proportional relationships go through the origin, (, ), and can be represented b the general equation = k. If the line of this proportional relationship intersects (, ), then = k. Therefore: k = The slope of the two lines is = + and the equation of the non-proportional line is: The line intersects the -ais at (, ). Another point on the line, based on the slope, can be found b going up units and to the right units from point (, ), which is point (, 8). 8 Copright Pearson Education, Inc. 8
Grade 8 Unit : Linear Relationships LESSON 7: REPRESENTATIONS 8.EE.. Table: D IV Graph: C C Table: C III Graph: B B Table: B II Graph: D D Table: A I Graph: A A 8.EE.. A Proportional relationship A has a smaller rate of change than B. 8.EE.. a. The slope of the line defined b the two points is: m = = = The equation of the line is = + b. If ou solve for b using the coordinates of one of the two points that the line intersects for eample, (, ) ou find that b =. The line is in fact a proportional relationship and the constant of proportionalit is equal to the slope: = m = b. To find whether the relationship is proportional, ou need to determine whether the line goes through the origin, (, ). If it does, then the relationship is proportional. If ou compare the sequence of values in the table, ou will notice that for each time the value of increases b, the value of increases b 7. Conversel, a decrease in the value of b corresponds to a decrease in the value of b 7. The first point in the table is (, 7). If ou remove from and 7 from, ou get (, ). Therefore, the line goes through the origin. Relationship B is a proportional relationship. The constant of proportionalit of this relationship is 7. c. As determined previousl, the constant of proportionalit for relationship A is and the constant of proportionalit for relationship B is 7. The constant of proportionalit for relationship C is. Since 7 is the greatest constant of proportionalit, relationship B has the greatest unit rate. Copright Pearson Education, Inc. 9
Grade 8 Unit : Linear Relationships LESSON 7: REPRESENTATIONS 8.EE.. A Relationship A B Relationship B A runner ran a half marathon at a constant rate of miles per hour. He started the race right on the starting line. D Relationship D A line is represented b the equation = m + b, where m = and b =. E Relationship E 8.EE.. Runner B Copright Pearson Education, Inc. 7
Grade 8 Unit : Linear Relationships LESSON 7: REPRESENTATIONS 8.EE. 8.EE.. = 8 The -intercept is. The slope can be calculated using an two points. For eample, using the points (, ) and (, ), the slope is ( ) =. The corresponding slope-intercept equation is =. 8.EE. 8.EE. 7. Time (wk) Height (cm) 7 9 Height of the Tomato Plant Over Time = + Height (cm) 8 Time (wk) 8 Copright Pearson Education, Inc. 7
Grade 8 Unit : Linear Relationships LESSON 7: REPRESENTATIONS 8.EE. 8. Here is an eample of a table. 8.. 7 9. = Challenge Problem 8.EE. 9. There are several possible was to find the point of intersection. B graphing You can graph the lines representing each equation in the same coordinate plane and estimate the coordinates of the intersection point. To verif the accurac of the values, ou can then use the equations to verif that the coordinates correctl solve the equations. = = 8( ) On the graph, the lines appear to intersect at (, ). If ou replace with in the equations, ou will find that: = () = = = 8( ) = 8() = The coordinates (, ) are correct. However, it is not eas to read the coordinates on the graph. Copright Pearson Education, Inc. 7
Grade 8 Unit : Linear Relationships LESSON 7: REPRESENTATIONS 8.EE. 9. B building data tables You can build a table of data points for each equation using the same -values, in the hopes of finding an -value that has the same -value in both tables. = 8 8 8 = 8( ) 8 8 When =, the -value is in both tables. Thus, the two lines intersect at this point. This method is eas to implement. In eas situations like this one, where the solution is a whole number, this method works well. However, imagine that = was not in the table. None of the points would have matched. You would have had to tr more values. B solving the equations At the point of intersection of the two lines, the value of is the same in both equations. Therefore: = 8( ) = 8 8 8 = 8 = = Once ou know, ou can solve for b replacing in one of the equations. = () = = The intersection point is (, ). Copright Pearson Education, Inc. 7
Grade 8 Unit : Linear Relationships LESSON 8: NEGATIVE SLOPE 8.EE.. A B E 8.EE.. C If m is positive and b is positive, the line will slope up from left to right. D If m is negative and b is positive, the line will slope down from left to right. E If m is positive and b is negative, the line will slope up from left to right. 8.EE.. Slope = 8.EE.. This relationship is not proportional because, even though the relationship is linear, it does not go through the origin. The line intersects the -ais at (, ), not at (, ). 8.EE.. Two points on line a are (, ) and (, ). The slope can be calculated as: m = = = Slope = The -intercept is the value of at the point of intersection of the line with the -ais. -intercept = Copright Pearson Education, Inc. 7
Grade 8 Unit : Linear Relationships LESSON 8: NEGATIVE SLOPE 8.EE.. Slope = -intercept = 8.EE. 7. 8.EE. 8. Slope = -intercept = 8 Slope = -intercept = 8.EE. 9. The slope of a line passing between two points is calculated using this formula: m = If the slope is negative, m is a negative number. So, if ( ) is negative then ( ) must be positive, and vice versa. A line with a negative slope alwas goes down as increases to the right. Here is a good eample of a line with slope. Take the two points (, ) and (, ) on the line. The -value of the second point () is greater than the -value of the first point (). It is the opposite for the -values: the -value of the second point ( ) is less than the -value of the first point (). As the -value of the line increases, the -value decreases. The line goes down from left to right. = 8.EE.. Equations A and B have a negative slope because the value of the slope m is a negative number in both equations. Copright Pearson Education, Inc. 7
Grade 8 Unit : Linear Relationships LESSON 8: NEGATIVE SLOPE 8.EE.. a. Maimum Heart Rate for Male Patients 8 = Maimum Heart Rate (bpm) 8 7 8 9 Age (r) b. -intercept = (, ) c. Slope = The unit of measurement for the slope is bpm/r. d. = = = 8 ( ) The -ear-old man s maimum heart rate is 8 bpm. 8.EE.. Answers will var based on age. Here is an eample for a -ear-old student who considers herself a good athlete. Running: 7 = bpm Rowing: 7 = bpm Biccling: 7 = 99 bpm Copright Pearson Education, Inc. 7
Grade 8 Unit : Linear Relationships LESSON 8: NEGATIVE SLOPE Challenge Problem 8.EE.. According to Rule, one of the equations has a -intercept of. The equation is therefore of the form = m +. Call this line. Rule states that both lines intersect at (, ), which means that this point is part of the line and the coordinates will solve the equation. For line : = m() + m = The equation of line is = +. The line has a negative slope (m is a negative number), which matches Rule. Rule sas that one of the equations represents a proportional relationship. Line does not represent a proportional relationship because it does not go through the origin (, ); instead, it crosses the -ais at the -intercept, (, ). Thus, the equation for line represents the proportional relationship. Proportional relationships follow the general formula = k. Since line also goes through (, ), ou can deduce the value of k b replacing and in the equation with the coordinates of the point. = k() k = The value of k also corresponds to the slope of the line. It is negative, which, once again, matches Rule. The two equations are: = + = = = + Copright Pearson Education, Inc. 77
Grade 8 Unit : Linear Relationships LESSON : THE EFFECT OF SCALE 8.EE.. B Slope = 8.EE.. D Slope = 8.EE.. C = + 8.EE.. A =. +. 8.EE.. Both graphs represent the same equation. The have the same slope and -intercept and contain all the same points. The different scaling of the aes makes the two graphs appear to have different slopes, though. Both lines have slope, and the both intersect the -ais at the point (, ). Therefore, the have the same slope-intercept equation: = +. If both lines are represented in coordinate planes with the same scale, the will look identical. 8.EE.. The line seems to intersect the -ais at (, ). Therefore, the value of the -intercept is. The line also appears to go through (., ). You can calculate the value of the slope using these points. m = = =.. Even though the slope looks like it might be, it is not. The graph is deceptive because the scales of the - and the -aes are not the same. The scale for the -ais is twice that of the scale for the -ais. The equation of the line is = +. Copright Pearson Education, Inc. 78
Grade 8 Unit : Linear Relationships LESSON : THE EFFECT OF SCALE 8.EE. 7. A B C.. 8.EE. 8. A B C D E = = + 7 = 7 = + = Copright Pearson Education, Inc. 79
Grade 8 Unit : Linear Relationships LESSON : THE EFFECT OF SCALE Challenge Problem 8.EE. 8.EE. 9. a. 7 (, ) (, ) line b (, ) line a = + b. The slope-intercept equation for line a is = +. The slope-intercept equation for line b is = +. c. (line a) (line b) (line c) d. The table compares the -values of various points on the three lines. Each -value of line c appears to represent the difference between the -value of line b and the -value of line a. c = b a You can confirm this observation b replacing the -values of lines a and b with their equivalent epressions using. = c b a = + + = + = + These calculations confirm that the equation of line c represents the difference between the other two equations. Copright Pearson Education, Inc. 8
Grade 8 Unit : Linear Relationships LESSON : PUTTING IT TOGETHER 8.EE. 8.EE.. Linear Linear and Proportional Non-Linear 8 7 7 8 7 8 7 8 8 7 7 8 8.EE. 8.EE.. Graphs With a Positive Constant Slope Graphs With a Negative Constant Slope Graphs With a Slope That is Not Constant 7 8 7 8 8 7 7 8 8 7 7 8 8.EE.. Slope =. or. 8.EE.. a. C Graph C b. B Graph B c. A Graph A 8.EE.. B Graph B C Graph C 8.EE.. Graph A: not possible (not a proportional relationship) Graph B: =.9 Graph C: =.89 Copright Pearson Education, Inc. 8
Grade 8 Unit : Linear Relationships LESSON : PUTTING IT TOGETHER 8.EE. 7. a. pounds = $.7 b. $.89 =. pounds Challenge Problem 8.EE. 8. The scale of the two aes will be ver different: the -ais must include much greater values than the -ais. The intervals for the -ais could be unit, but the intervals for the -ais should be something like, units. Conversion From Feet to Miles Number of Miles,,,,,, Number of Feet Copright Pearson Education, Inc. 8
Grade 8 Unit : Linear Relationships LESSON : PUTTING IT TOGETHER 8.EE., 8.EE.. Definitions and eamples will var. Here are some eamples. Word or Phrase nonproportional linear relationship slope Definition A linear relationship between two variables that does not go through the origin, (, ) This tpe of relationship can be represented with the general equation = m + b, where m is the slope of the line and b is the -intercept. This equation is known as slopeintercept form. The inclination of the graph of a line In the slope-intercept form of a linear equation, = m + b, m represents the slope. For an two distinct points on the line, such as (, ) and (, ), the value of the slope is: m = Eamples (, ) = + m = b = Line = + has a slope of. (, ) = + m = b = If the slope is positive, the line will slope up from left to right. If the slope is negative, the line will slope down from left to right. The line passing through points A(, ) and B(, ) has a slope of : m = ( ) = = + A B slope = Copright Pearson Education, Inc. 8
Grade 8 Unit : Linear Relationships LESSON : PUTTING IT TOGETHER 8.EE., 8.EE.. Word or Phrase Definition Eamples -intercept The value of at the point of intersection between a line and the -ais Line = + has a -intercept of. = + In the slope-intercept form of a linear equation, = m + b, b represents the -intercept. (, ) b = Line = has a -intercept of. = (, ) b = Copright Pearson Education, Inc. 8
Grade 8 Unit : Linear Relationships LESSON : PUTTING IT TOGETHER 8.EE., 8.EE.. Here is one eample. An amusement park has two formulas for the pricing of rides. Formula A: To get into the amusement park, ou pa a $ entrance fee. Once ou are in the park, each ride costs $. Formula B: There is no entrance fee, but each ride costs $. If ou want to represent the total cost of a da at the amusement park depending on the number of rides, ou can use as the total cost and as the number of rides. Formula A is not a proportional relationship because there is a minimum cost (flat fee) of $. This means that even if ou do not go on an rides, ou still pa $. The equation of this formula is = +. Formula B is a proportional relationship because the cost is directl proportional to the number of rides ou go on. The equation of this formula is =. Total Cost of Rides ($) Amusement Park Cost per Da (Based on the Number of Rides) = (, ) = + Number of Rides In this situation, the lines intersect at (, ), which means that with rides, the cost is the same with both formulas. With fewer than rides, formula B is cheaper. If ou go on more than rides, formula A is cheaper. Copright Pearson Education, Inc. 8