GRAPH MATHEMATICS (SAMACHEERKALVI) HARDWORK IS THE BEST WEAPON TO DEFEAT FAILURE. SELF CONFIDENCE +HARDWORK = SUCCESS X- STANDARD

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1 1 GRAPH (Fully solved for better understanding and to work out easily) MATHEMATICS (SAMACHEERKALVI) HARDWORK IS THE BEST WEAPON TO DEFEAT FAILURE SELF CONFIDENCE +HARDWORK = SUCCESS X- STANDARD PREPARED BY, A.HAJA MOHAIDHEEN, M.Sc., B.Ed., G.H.S, KONUR, VILLUPURAM CELL:

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3 Exercise Draw the graph of the following functions. (i) y = 3x 2 Table for y = 3x 2 3 x x y Ordered pair of points : (-3, 27), (-2, 12), (-1, 3), (0, 0), (1, 3), (2, 12), (3, 27). Plot the points on the graph sheet and join them by a smooth curve Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 2 units This parabola is the graph of y = 3x 2 This is an open upward parabola

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5 5 1. (ii). y = - 4 x 2 Table for y = - 4 x 2 x x y Ordered pair of points: (-3, -36), (-2, -16), (-1, -4), (0, 0), (1, -4), (2, -16), (3, -36). Plot the points on the graph sheet and join them by a smooth curve Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 2 units This parabola is the graph of y = -4x 2 This is an open downward parabola

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7 7 1.(iii). y = ( x+2) (x +4) Table for y = ( x+2) (x +4) x x x y Here product Ordered pair of points: (-7, 15), (-6, 8), (-5, 3), (-4, 0), (-3, -1), (-2,0), (-1, 3), (0, 8), (1, 15). Plot the points on the graph sheet and join them by a smooth curve Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 1 unit This parabola is the graph of y = ( x+2) (x +4) This is an open upward parabola

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9 9 1. (iv). y = 2 x 2 -x + 3 Table for y = 2 x 2 -x + 3 x x x x y Ordered pair of points: (-3, 24), (-2, 13), (-1, 6), ( 0, 3), (1, 4), (2, 9), (3, 18). Plot the points on the graph sheet and join them by a smooth curve Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 2 units This parabola is the graph of y = 2 x 2 -x + 3 This is an open upward parabola

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11 2. Solve the following equations graphically. (TYPE-I) (i). x 2-4 =0 Let y = x 2-4 Table for y = x 2-4: 11 x x y Ordered pair of points: (-4, 12), (-3, 5), (-2, 0), (-1, -3), ( 0, -4), (1, -3), (2, 0), (3, 5), (4, 12). Plot the points on the graph sheet and join them by a smooth curve Solving y = x 2-4 Note: ௦ = x The corresponding y values (-) (-) ( + ) of the solution set in the table are 0 y = 0 ( x- axis) 2. If the coefficient of x 2 is 1, then the values of y are repeated Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 1 unit This parabola cuts the X axis at the points (-2, 0) and (2, 0). Solution set: x= { -2, 2}

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13 13 2.(ii). x 2-3x-10=0 Let y = x 2-3x-10 Table for y = x 2-3x-10: x x x y Ordered pair of points: (-3, 8), (-2, 0), (-1, -6), (0, -10), (1, -12), (2, -12), (3, -10), (4, -6), (5, 0), (6, 8). Plot the points on the graph sheet and join them by a smooth curve Solving y = x 2-3x ௦ = x 2-3x -10 (-) (-) ( + ) ( + ) y = 0 ( x- axis) Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 1 unit This parabola cuts the X axis at the points (-2, 0) and (5, 0). Solution set: x= { -2, 5} Soln:- x= -2 & x= +5 The table should contain these limits(values) Note: 1. The corresponding y values of the solution set in the table are 0 2. If the coefficient of x 2 is 1, then the values of y are repeated

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15 2.(iii). (x-5)(x-1) =0 Let y= (x-5)(x-1) Table for y= (x-5)(x-1) 15 Rough work: (x-5)(x-1)=0 x-5=0 or x-1=0 x=5 or x=1 The table should contain these limits(values) x x x y Here product Ordered pair of points: (-1, 12), (0, 5), (1, 0), (2, -3), (3, -4), (4, -3), (5, 0), (6, 5), (7, 12). Plot the points on the graph sheet and join them by a smooth curve Solving y = (x-5)(x-1) 0 = (x-5)(x-1) (-) (-) y = 0 ( x- axis) Scale : Hint: If we take the limits from 0 to 6, the graph will be in small size. Take the limits from 1 to 7 x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 1 unit This parabola cuts the X axis at the points (1, 0) and (5, 0) Solution set: x= { 1, 5}

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17 2.(iv). (2x +1)(x-3)=0 Let y= (2x +1)(x-3) Table for y= (2x +1)(x-3) 17 Rough work: (2x+1)(x-3)=0 2x+1=0 or x-3=0 2x=-1 or x=3 x= -1/2 The table should contain these limits(values) x x x x y Here product Ordered pair of points: (-2, 15), (-1, 4), (0, -3), (1, -6), (2, -5), (3, 0), (4, 9). Plot the points on the graph sheet and join them by a smooth curve Solving y = (2x +1)(x-3) 0 = (2x +1)(x-3) (-) (-) y = 0 ( x- axis) Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 1 unit This parabola cuts the X axis at the points (-0.5, 0) and (3, 0) Solution set: x= { -0.5, 3}

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19 3. Draw the graph of y= x 2 and hence solve x 2-4x-5=0. (TYPE-II) Table for y= x 2 : x y Ordered pair of points: (-5, 25), (-4, 16), (-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9), (4, 16), (5, 25). Plot the points on the graph sheet and join them by a smooth curve Solving y = x 2 0 = x 2-4x (-) (-) ( + ) ( + ) y = 4x + 5 (Solution line) Table for y = 4x +5 x x y Ordered pair of points: (-2, -3), (-1, 1), (0, 5), (1, 9), (2, 13). Plot the points on the graph sheet and join them by a straight line. Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 2 units This straight y = 4x + 5 line cuts the parabola at x = -1 and x = 5. Solution set: x= { -1, 5} Soln:- x= -1 & x= +5 The table should contain these limits(values)

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21 21 4. Draw the graph of y =x 2 +2x-3 and hence find the roots of x 2 - x- 6= 0.(TYPE-II) Table for y = x 2 +2x-3 x x x y Ordered pair of points: (-4, 5), (-3, 0), (-2, -3), (-1, -4), (0, -3), (1, 0), (2, 5), (3, 12). Plot the points on the graph sheet and join them by a smooth curve Solving y =x 2 +2x = x 2 x - 6 Soln:- x= -2 & x= +3 The table should contain these (-) (-) ( + ) ( + ) limits(values) y = 3x +3 (Solution line) Table for y = 3x +3 x x y Ordered pair of points: (-2, -3), (-1, 0), (0, 3), (1, 6), (2, 9), Plot the points on the graph sheet and join them by a straight line. Scale :: x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 1 unit This straight y = 3x +3 line cuts the parabola at x = -2 and x = 3 Solution set: x= { -2, 3}

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23 23 5. Draw the graph of y = 2x 2 + x - 6 and hence solve 2x 2 + x - 10= 0. (TYPE-III) Table for y = 2x 2 + x - 6 x x x x y Ordered pair of points: (-3, 9), (-2, 0), (-1, -5), (0, -6), (1, -3), (2, 4), (3, 15). Plot the points on the graph sheet and join them by a smooth curve Solving y =2x 2 + x = 2x 2 + x - 10 Soln:- x= -5/2 x= +4/2 x= ; x=2 (-) (-) ( - ) ( + ) The table should contain these limits(values) y = 4 (Solution line) Table for y = 4 x y Ordered pair of points: (-2, 4), (-1, 4), (0, 4), (1, 4), (2, 4), Plot the points on the graph sheet and join them by a straight line. Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 1 unit This straight y = 4 line cuts the parabola at x = -2.5 and x = 2 Solution set: x= { -2.5, 2}

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25 6. Draw the graph of y = x 2 -x - 8 and hence find the roots of x 2-2 x - 15= 0. (TYPE-II) Table for y = x 2 -x - 8: 25 x x x y Ordered pair of points: (-4, 12), (-3, 4), (-2, -2), (-1, -6), (0, -8), (1, -8), (2, -6), (3, -2), (4, 4), (5, 12). Plot the points on the graph sheet and join them by a smooth curve Solving y = x 2 -x = x 2-2 x - 15 Soln:- x= -3 & x= +5 The table should contain these (-) (-) ( + ) ( + ) limits(values) y = x + 7 (Solution line) Table for y = x + 7 : x y Ordered pair of points: (-2, 5), (-1, 6), (0, 7), (1, 8), (2, 9), Plot the points on the graph sheet and join them by a straight line. Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 1 unit This straight y = x + 7 line cuts the parabola at x = -3 and x = 5 Solution set: x= { -3, 5}

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27 27 7. Draw the graph of y = x 2 + x -12 and hence solve x 2 +2 x + 2= 0. (TYPE-IV) Table for y= x 2 + x -12 x x x y Ordered pair of points: (-4, 0), (-3, -6), (-2, -10), (-1, -12), (0, -12), (1, -10), (2, -6), (3, -0). Plot the points on the graph sheet and join them by a smooth curve Solving y = x 2 Roughwork: +x -12 x 2 +2 x + 2= 0 can t be 0 = x 2 +2 x + 2 factorized No solution (-) (-) ( - ) ( - ) y = - x - 14 (Solution line) Table for y = - x 14 x x y Ordered pair of points: (-2, -12), (-1, -13), (0, -14), (1, -15), (2, -16), Plot the points on the graph sheet and join them by a straight line. Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 1 unit This straight y = - x - 14 line does not cut the parabola at anywhere. There is no real solution for x 2 +2 x + 2= 0.

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29 29 Some Special Graphs (Exercise 10.2) 1. A bus travels at a speed of 40 km / hr. Write the distance-time formula and draw the graph of it. Hence, find the distance travelled in 3 hours. Solution: Distance = Speed Time Let d = 40t. ( d Distance ; t Time) x (Time) y (Distance) From the table, we found that as x increases, y also increases. This is a direct variation. Equation for direct variation : y= kx = k= y/x, k is the constant of proportionality. k = = = = = = 40 y = 40x forms a straight line graph. Ordered pair of points: (1, 40), (2, 80), (3, 120), (4, 160), (5, 200), Plot the points on the graph sheet and join them by a straight line Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 40 units This line is the graph of y = 40x From the graph, we find that, The distance travelled in 3 hours is 120 km

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31 2. The following table gives the cost and number of notebooks bought. 31 x y Draw the graph and hence (i) Find the cost of seven note books. (ii) How many note books can be bought for Rs Solution: x y From the table, we found that as x increases, y also increases. This is a direct variation. Equation for direct variation : y= kx = k= y/x, k is the constant of proportionality. k = 3 = = = = = = 15 y = 15x forms a straight line graph. Ordered pair of points: (2, 30), (4, 60), (6, 90), (8, 120), (10, 150), (12, 180), Plot the points on the graph sheet and join them by a straight line Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 15 units This line is the graph of y = 15x From the graph, we find that, (i). The cost of seven note books = Rs. 105 (ii). No. of note books can be bought for Rs. 165 = 11.

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33 33 3. x y Draw the graph for the above table and hence find (i) the value of y if x = 4 (ii) the value of x if y = 12 Solution: x y From the table, we found that as x increases, y also increases. This is a direct variation. Equation for direct variation : y= kx = k= y/x, k is the constant of proportionality. k = 2 = = = = = 2 y = 2x forms a straight line graph. Ordered pair of points: (1, 2), (3, 6), (5, 10), (7, 14), (8, 16), ), Plot the points on the graph sheet and join them by a straight line Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 2 units This line is the graph of y = 2x From the graph, we find that, (i). When x=4, then y = 8 (ii). When y=12 then x = 6

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35 35 4. The cost of the milk per litre is Rs.15. Draw the graph for the relation between the quantity and cost. Hence find (i) the proportionality constant (ii) the cost of 3 litres of milk. Solution: No. of litres x Cost of milk y From the table, we found that as x increases, y also increases. This is a direct variation. Equation for direct variation: y= kx = k= y/x, k is the constant of proportionality. k = = = = = = 15 y = 15x forms a straight line graph. Ordered pair of points: (1, 15), (2, 30), (3, 45), (4, 60), (5, 75), Plot the points on the graph sheet and join them by a straight line Scale : x axis: 1 c.m. = 1 unit y axis: 1 c.m. = 15 units This line is the graph of y = 15x From the table and graph, we find that, (i). The proportionality constant = 15 (ii). The cost of 3 litres of milk. = Rs. 45

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37 37 4. Draw the Graph of xy = 20, x, y >0. Use the graph to find y when x =5, and to find x when y =10 Solution:: xy = 20 ; y = 20/x x y From the table, we found that as x increases, y decreases. This is an indirect variation. Equation for indirect variation: xy= k k = (1x 20)= (2x 10)= (4x 5)= (5x 4)= (10x 2) )= (20x 1)= 20 xy = 20 forms a rectangular hyperbola. Ordered pair of points: (1, 20), (2, 10), (4, 5), (5, 4), (10, 2), (20, 1). Plot the points on the graph sheet and join them by a smooth curve Scale : x axis: 1 c.m. = 2 units y axis: 1 c.m. = 2 units This curve is the graph of xy = 20. From the graph, we find that, (i). When x =5 then y = 4 (ii). When y =10 then x = 2

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39 39 5. No. of workers x No. of days y Draw graph for the data given in the table. Hence find the number of days taken by 12 workers to complete the work. Solution: x y From the table, we found that as x increases, y decreases. This is an indirect variation. Equation for indirect variation: xy= k k = (3x 96)= (4x 72)= (6x 48)= (8x 36)= (9x 32) )= (16x 18)= 288 xy = 288 forms a rectangular hyperbola. Ordered pair of points: (3, 96), (4, 72), (6, 48), (8, 36), (9, 32), (16, 18). Plot the points on the graph sheet and join them by a smooth curve Scale : x axis: 1 c.m. = 2 units y axis: 1 c.m. = 10 units This curve is the graph of xy = 288 From the graph, we find that, the number of days taken by 12 workers to complete the work = 24 days ************ ALL THE BEST *********** PREPARED BY, A.HAJA MOHAIDHEEN, M.Sc., B.Ed., G.H.S, KONUR, VILLUPURAM CELL:

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