Q1-Mathematics-8-Module-4 PDF

8 Mathematics Quarter 1 – Module 4: Graphing Linear Equations and Describing the Graph of Linear Equations Mathematics – Grade 8 Quarter 1 – Module 4: Graphing Linear Equations and Describing the Graph of Linear Equations First Edition, 2020 Republic Act 8293, section 176...

8 Mathematics Quarter 1 – Module 4: Graphing Linear Equations and Describing the Graph of Linear Equations Mathematics – Grade 8 Quarter 1 – Module 4: Graphing Linear Equations and Describing the Graph of Linear Equations First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education – Region XI Regional Director: Allan G. Farnazo, CESO IV Assistant Regional Director: Maria Ines C. Asuncion, EdD, CESO V Development Team of the Module Writer: Larry C. Cezar Contributor: Maria Fe P. Ursal Editor: Donna Marie M. Señedo Reviewer: Donna Marie M. Señedo Illustrator: Larry C. Cezar Layout Artist: Template Developer: Management Team: Reynaldo M. Guillena, CESO V Jinky B. Firman, PhD, CESE Marilyn V. Deduyo Alma C. Cifra, EdD Aris B. Juanillo, PhD May Ann M. Jumuad, PhD Antonio A. Apat Printed in the Philippines by Davao City Division Learning Resources Management Development System (LRMDS) Department of Education – Region XI Davao City Division Office Address: DepEd Davao City Division, E. Quirino Ave., Davao City, Davao del Sur, Philippines Telefax: (082) 224-0100 E-mail Address: [email protected]  4 Mathematics Quarter 1 – Module 4: Graphing Linear Equations and Describing the Graph of Linear Equations M8AL – If – 2; M8AL – If – 3 Introductory Message For the facilitator: As a facilitator, you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning at home. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module. For the learner: As a learner, you must learn to become responsible of your own learning. Take time to read, understand, and perform the different activities in the module. As you go through the different activities of this module be reminded of the following: 1. Use the module with care. Do not put unnecessary mark/s on any part of the module. Use a separate sheet of paper in answering the exercises. 2. Don’t forget to answer Let Us Try before moving on to the other activities. 3. Read the instructions carefully before doing each task. 4. Observe honesty and integrity in doing the tasks and checking your answers. 5. Finish the task at hand before proceeding to the next. 6. Return this module to your teacher/facilitator once you are done. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it! ii  Let Us Learn This module was designed and written with you in mind. It will help you understand better on how to graph linear equations given any two points, the x- and y- intercepts, and the slope and a point on the line. The scope of this module permits it to be used in many different learning situations. The lessons are done to follow the standard sequence of the course. In this module you are expected to: graph linear equations in two variables given: a. graph linear equations given any two points; b. graph linear equations given the x- and the y-intercepts; c. graph linear equations given the slope and a point on the line; and d. describe the graph of linear equations Let Us Try Let us have a quick review. Consider the coordinate plane below. 1. Can you plot the ordered pairs (1, 4) and (3, 2) in a rectangular coordinate system? 2. If you are going to connect the two points, can you form a straight line? 3. Will the given ordered pairs satisfy the equation x + y = 5? 1  Let Us Study In this module, you will learn about the graphing of linear equations given two points, the x- and y-intercepts, and using the slope and a point on the line, and to describe the graph of linear equations. To guide you better, you will have to perform the simple activities below. Activity 1: Let’s Explore! Direction: Plot the given ordered pairs in a rectangular coordinate system then connect the points consecutively (A-B-C-D-E-A): Point A (4, 3) is already done for you A (4, 3) A B (-3, 2) C (3, -1) D (0, 5) E (-2, -2) After connecting the points consecutively, what was the figure formed? Activity 2: Identify the coordinates of the points of each graph. 1. ( , ) and ( , ) 2. ( , ) and ( , ) 3. ( , ) and ( , ) Were you able to finished the task? If yes, you may now proceed to the next part of the module. If no, take time to complete the task for you to better understand the next discussion. 2 Let us examine the equation x + 2y = 4. When a certain number is substituted to one of the variables in the equation, it will give a pair of coordinates; example (2, 1) and (-2, 3). Can we graph the equation with these pair of coordinates? We have three different ways of graphing an equation. These are: (a) given any two points; (b) given the x- and the y- intercepts and (c) given the slope and a point on the line. A. Graphing linear equations given any two points To graph linear equations, 1: Make a table of values. Using the given equation, assign values for x to solve for y. 2: Plot at least two (2) ordered pairs in the cartesian plane. 3: Connect the points to form a straight line. Example 1. Graph the equation x + 2y = 4.. Solution: Make a table of values. Assign values for x to solve for y. To solve for y, substitute the chosen value of x to the given equation If x = 0 If x = 2 𝑥 + 2𝑦 = 4 𝑥 + 2𝑦 = 4 0 + 2𝑦 = 4 2 + 2𝑦 = 4 2𝑦 4 2𝑦 = 4 − 2 = 2 2 2𝑦 2 𝑦 =2 = 2 2 𝑦 =1 Write the values of x and y in the table. X y (x, y) 0 2 (0, 2) 2 1 (2, 1) Take note, to graph the equation we need at least two points. Plot the ordered pairs (0, 2) and (2, 1) in the rectangular coordinate system (Figure 1) and connect the points to form a straight line (Figure 2). 3  (0,2 ) (2, 1) Figure 1 Figure 2 The graph of the linear equation is a straight line (Figure 2). Example 2. Graph the equation 3x – y = 6 Solution: If x = 0 If x = 1 3𝑥 − 𝑦 = 6 3𝑥 − 𝑦 = 6 3(0) − 𝑦 = 6 3(1) − 𝑦 = 6 0−𝑦 =6 3−𝑦 =6 −1 ( −𝑦) = (6)(−1) −𝑦 = 6 − 3 𝑦 = −6 (−1)(− 𝑦) = (3)(−1) 𝑦 = −3 Write the values of x and y in the table. Using the ordered pairs from the table of values, the graph will be: x y (x, y) 0 -6 (0, -6) 1 -3 (1, -3) 4 B. Graphing linear equations using the x-and the y- intercepts Intercept is defined as the point where a graph crosses the x or y axis. The point (0, y) where the graph crosses the y-axis is the y-intercept and the point (x, 0) where the graph crosses the x-axis is the x-intercept. Now, we will use the same equation x + 2y = 4 given in example 1 and graph this linear equation using the x- and the y-intercepts. Solution: a. To find the x-intercept, let y = 0. b. To find the y-intercept, let x = 0 x + 2y = 4 x + 2y = 4 x + 2(0) = 4 0 + 2y = 4 x+0=4 2y = 4 2𝑦 4 x=4 2 = 2 y=2 x-intercept: (4, 0) y-intercept: (0, 2) Plot the x- and y-intercepts and connect the points to draw a straight line. y-intercept x-intercept The straight line that has been drawn is the graph of the equation x + 2y = 4. 5 Example 2. Graph the equation 2x - 3y = 6 using the x- and the y-intercepts. Solution: a. To find the x-intercept, let y = 0. b. To find the y-intercept, let x = 0 2x - 3y = 6 2x - 3y = 6 2x – 3(0) = 6 2(0) - 3y = 6 2x – 0 = 6 0 - 3y = 6 2𝑥 6 −3𝑦 6 = = 2 2 −3 −3 x=3 y = -2 x-intercept: (3, 0) y-intercept: (0, -2) Therefore, the x-intercept is (3, 0) and y-intercept is (0, -2). The graph of the equation 2x - 3y = 6 will be: Observing the graphs of the given two equations, the line crosses the x-axis where y=0 and crosses the y-axis where x = 0. These points are the x- and the y- intercepts. 6 C. Graphing a Linear Equation Using the Slope and a Point on the Line The third method that we are going to use in graphing an equation is using the slope of the equation and a point on the line. In this method, still, we will be using same equation x + 2y = 4 just like in our first and second method. To graph using the slope and a point on the line, we will do the following steps: STEP 1: Determine if the given equation is in standard form, transform the equation into slope-intercept form, y = mx + b Since the equation x + 2y = 4 is in standard form, we are going to transform this equation into slope-intercept form y = mx + b, that is x + 2y = 4 2y = 4 – x - subtraction property of equality Or 2y = -x + 4 2𝑦 −𝑥 4 2 = 2 + 2 - divide both sides by 2 𝑥 1 y=- +2 or y =- x+2 2 2 Take note, in the slope intercept form y = mx + b, m is the slope and b is the y- 1 𝟏 intercept, therefore, in our equation y =- 2 x + 2, the slope (m) is - 𝟐 and the y- intercept (b) is 2. Since the y- intercept is a point on the line, we will use it as our reference point. These values now will be used in graphing the equation x + 2y = 4. STEP 2: Locate the y-intercept (0, 2) in the y-intercept coordinate plane and plot the point (see Figure A) 1 Using the slope, - , apply the definition Figure A 2 𝑟𝑖𝑠𝑒 1 m= 𝑟𝑢𝑛 = - 2 , where we go down 1 unit since it is negative and run 2 units to the right. Or rise 1 unit, and since it is negative run 2 units to the left. (Either way we still get the same result). 7 STEP 3: From the y-intercept (0,2), move one (1) unit downward (since rise is negative), then run 2 units going to the right, and mark the second point. Observe that the two points now has an rise ordered pair (0, 2) and (2,1). (See Figure run B) Figure B STEP 4: Connect the points by drawing a straight line passing through them. This is now the graph of the equation x + 2y = 4 using the slope and a point on the line. (See Figure C) Figure C The given graph below is the graph of the equation x + 2y = 4. Take note that four points are plotted on the graph and all of the points lie on the same straight line. These points are ordered pairs that satisfy the equation. These four sets of ordered pairs are taken from the three methods that we have used in graphing the equation. This only shows that using different methods in graphing an equation will still result to the same straight line. Figure D 8 Describing the Graph of an Equation Identifying the slope of an equation using the slope-intercept form can be our guide in describing the graph of any equation. Example: Graph each equation using its slope and y-intercept: 4 a) y = 3 x–1 c) y = -3 b) x + 2y = 6 d) x = 5 Solution: 𝟒 a) y = 𝟑 x–1 The slope is and its y-intercept is -1, whose coordinates are (0, -1). To graph the equation, start by plotting the y-intercept which is (0, -1). 4 𝑟𝑖𝑠𝑒 Next, use the slope(m)= 3 = 𝑟𝑢𝑛 which means from the y-intercept (0, -1), rise 4 units then run 3 units to the right since the slope is a positive, which leads us to point (3, 3). 4 Since the slope is 3 positive, so the trend of the graph is increasing from left to right. (0, -1) begin b) x + 2y = 6 Transform the equation into slope-intercept form. (y = mx + b) x + 2y = 6 2y = -x + 6 - Subtraction Property of Equality 2𝑦 1 6 2 = - 2x + 2 - Division Property of Equality 1 y=- 2 x +3 - Slope-Intercept Form 1 The slope is - 2 and the y-intercept is 3, whose coordinates are (0, 3). To graph 1 the equation, we begin by plotting the y-intercept (0, 3) and using the slope = − 2, from the point (0,3) we rise 1 unit and run 2 units to the left since our slope is a negative, which lead us to point (-2, 4). Another way to do it is to begin at the y - intercept (0, 3) then, go down 1 unit since the slope is negative and run 2 units to the right, which leads us to point (2, 2). Lastly, connect the points, to form a straight line. 9  (-2, 4) 1 (0, 3) begin Since the slope - 2 is (2, 2) negative, so the trend of the graph is decreasing from left to right. c) y = -3 The equation has a y-intercept only which is -3, whose coordinates are (0, -3). To graph the equation, start with the y-intercept (0, -3). Next, since there has no slope, so the graph will be a plain horizonal line. Since the slope is zero (0), so the trend of the graph is a (0, -3) horizontal line. d) x = 5 The graph of this equation is a vertical line passing through the x axis at point (5,0). The slope is undefined so the (5, 0) trend of the graph is a vertical line. 10  Let Us Practice Now, it’s your turn to apply the concepts you have learned on graphing linear equations given any two points, given the x- and y-intercepts, and given the slope and a point on the line. Activity: “Straight Me Up!” DIRECTION: Graph each of the given linear equations. You can use any of the methods mentioned above. 1) x + 2y = 4 2) 2x - y = 2 3) y = 2x +1 4) x + y = -3 5) 3x + y = 3 6) 3x - 2y = 6 11  Let Us Practice More Activity: “Complete Me!” DIRECTION: Complete the table and graph each equation using the ordered pairs. 1) 3x – y = 3 ordered pair x y (x, y) 0 1 2 2) x + y = 2 ordered pair x y (x, y) -2 0 2 3) y = 3x - 1 ordered pair x y (x, y) -1 0 1 12  Let Us Remember The graph of all solutions of an equation is a straight line of infinite length as indicated by the arrows on both ends. Remember, that there are three methods that we can use in graphing linear equations. Given any two points Given the x- and y-intercepts Given the slope and a point on the line Important Ideas on Slopes A line with a positive slope A line with a negative slope increases from left to right. decreases from left to right. A horizontal line has 0 slope. A vertical line has an undefined slope. Let Us Assess Multiple Choice. Read and analyze the following items and choose the letter of the correct answer from the given choices. Write the letter of the correct answer on a separate sheet of paper. 1. A line represents the graph of __________. a. quadratic equation c. monomial b. polynomials d. linear equation 2. The __________ of the line is the value of x when y = 0. a. y-intercept c. slope b. x-intercept d. none of the above 3. In graphing linear equations in two variables using slope and y-intercept form, if m > 0, the graph is a line that rises from _______ to _______. a. right, left c. left, right b. Top, bottom d. none of the above 13 4. What is the trend of the graph of a linear equation whose slope is negative? a. vertical line c. horizontal line b. rises from left to right d. decreases from left to right 5. Which of the following points passes through 3x + y = -3? a. (0, 3) c. (1, 3) b. (-1, 0) d. (-3, 0) 6. If the equation y = 3x - 2 will be graphed using the y-intercept and its slope, the first point to be located on the graph will be a. (0, 2) c. (0, -2) b. (0, 3) d. (3, 0) 7. What is the trend of the graph of the equation y = 3x + 5? a. increases from left to right c. a vertical line b. falls from left to right d. a horizontal line 8. What is the slope of the equation x - 3y = 6? a. 3 c. -3 1 −1 b. d. 3 3 9. Which is the graph of the equation 3x + 2y = 6? a. c. b. d. 10. Which of the following is the equation of the graph below? a) y = x + 1 b) y = -x + 1 c) y = x – 1 d) y = -x - 1 14  Let Us Enhance Activity: “Light Me Up!” DIRECTION: Consider the situation below and answer the questions that follow. Situation: Criz made an experiment by lighting an 8 inches long candle. Then, he recorded the length of the candle represented by x (time in hours) and y (length of the candle). Time in Hours (x) 0 1 2 Length of the candle (y) 8 6 4 1. If you complete the table, what will be the length of the candle after 3 hours? ____________________________________________________________ 2. Based on the observation of Criz, considering the rate the candle melted, how long (in hours) will the candle be used up completely? ____________________________________________________________ 3. If you graph the result of the experiment, what will be the trend of the graph? ____________________________________________________________ Let Us Reflect Heart and Think Moments Write three (3) things you have learned about the lesson ____________________________________________ ____________________________________________ ____________________________________________ Write 1 question you would like to ask about the topic ____________________________________________ 15 Answer Key 1. 16  17 Let Us Reflect Answer may vary. Let Us Enhance 1. 2 inches 2. 4 hours 3. The graph decreases from left to right. Let Us Assess 1. D 6. C 2. B 7. A 3. C 8. B 4. D 9. A 5. B 10. C  References Abuzo, Emmanuel, Merden Bryant, Jem Boy Cabrella, Belen Caldez, Melvin Callanta, Anastacia Proserfina Castro, Alicia Halabaso, Sonia Javier, Roger Nocom, and Concepcion Ternida. Mathematics Learner's Module 8. 1st ed. Reprint, Department of Education, 2013. "Graphs of Linear Equation In Two Variables - Module 2". Word, 2014. 6485. DepEd Learning Portal. Graphing Lines In Slope-Intercept Form. Ebook. Kuta Software. Accessed 19 May 2020.https://www.rcsdk12.org/cms/lib/NY01001156/Centricity/Dom ain/4553/G raphing%20Lines%20in%20Slope-Intercept%20Form.pdf. Graphing Using Intercepts. Ebook. Accessed 19 May 2020. https://www.anderson5.net/cms/lib02/SC01001931/Centricity/Dom ain/2147/G raphing%20using%20x%20and%20y-intercepts.pdf Graphing Using Intercepts Worksheet. Ebook. Jensen. Accessed 19 May 2020. https://www.jensenmath.ca/6.3%20worksheet-2.pdf 18 For inquiries or feedback, please write or call: Department of Education – Region XI F. Torres St., Davao City Telefax: Email Address: [email protected]