13 Graphs, Equations and Inequalities

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1 13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line, a small solid circle is used for or and a hollow circle is used for > or <. For eample, Here the solid circle means that the value 5 is included. < Here the hollow circle means that the value 7 is not included. When solving linear inequalities we use the same techniques as those used for solving linear equations. The important eception to this is that when multipling or dividing b a negative number, ou must reverse the direction of the inequalit. However, in practice, it is best to tr to avoid doing this. Eample 1 Solve the inequalit + 6 > 3 and illustrate the solution on a number line. + 6 > 3 > 3 6 > 3 Subtracting 6 from both sides of the inequalit This can be illustrated as shown below:

2 13.1 Eample Solve the inequalit and illustrate the solution on a number line Subtracting 7 from both sides Dividing both sides b 3 This can now be shown on a number line Eample 3 Illustrate the solution to the inequalit Because this inequalit contains the term ' 3', first add 3 to both sides to remove the sign Adding 3 to both sides 6 3 Subtracting 6 from both sides or Dividing both sides b 3 This is illustrated below. Eample Solve the equation 7 < In an inequalit of this tpe ou must appl the same operation to each of the 3 parts. 7 < < 5 0 Subtracting 3 from both sides < Dividing both sides b 5 This can then be illustrated as below

3 Eercises 1. Draw diagrams to illustrate the following inequalities: (a) > 3 (c) (d) 3 (e) < (f) 0 3. Write down the inequalit represented b each of the following diagrams: (a) (c) (d) (e) Solve each of the following inequalities and illustrate the results on a number line. (a) + 7 > 1 6 > 3 (c) 0 (d) 5 10 (e) (f) 6 3 (g) (h) 3 (i) 1. Solve each of the following inequalities and illustrate our solutions on a number line. (a) > 7 (c) 3 9 < 6 (d) + 30 (e) (f) + 1 > (g) + > 3 (h) (i) Solve the following inequalities, illustrating our solutions on a number line. (a) < (c) (d) 1 18 (e) 0 8 < (f) 3 9 (g) (h) (i) 7 <

4 Given that the perimeter of the rectangle shown is less than, form and solve an inequalit The perimeter of the triangle shown is greater than 1 but less than or equal to 30. Form and solve an inequalit using this information The area of the rectangle shown must be less than 50 but greater than or equal to 10. Form and solve an inequalit for A cclist travels at a constant speed v miles per hour. He travels 30 miles in a time that is greater than 3 hours but less than 5 hours. Form an inequalit for v. 10. The area of a circle must be greater than or equal to 10 m and less than 0 m. Determine an inequalit that the radius, r, of the circle must satisf. 11. The pattern shown is formed b straight lines of equations in the first quadrant. 3 =1 = = 3 = = 3 (a) One region of the pattern can be described b the inequalities Cop the diagram and put an R in the single region of the pattern that is described. 10

5 This is another pattern formed b straight line graphs of equations in the first quadrant. = = = = The shaded region can be described b three inequalities. = Write down these three inequalities. 0 (KS3/96/Ma/Tier 6-8/P1) 13. Graphs of Quadratic Functions In this section we recap the graphs of straight lines before looking at the graphs of quadratic functions. A straight line has equation = m + c where m is the gradient and c is the -intercept. Eample 1 (a) Draw the lines with equations, = + 8 and = + 3. Describe the translation that would move = + 8 onto = + 3. (a) To plot the graphs, we calculate the coordinates of three points on each line. For For = + 8 we have ( 3, 5), (0, 8) and (, 1). = + 3 we have, (, 1), (0, 3) and (3, 6). The graphs are shown below. 103

6 = + 8 = A translation along the vector the line = would move the line = + 8 onto 5 Eample Draw the curve with equation =. = is not a linear equation, so we will have to draw a smooth curve. To do this we need to calculate and plot a reasonable number of points. We begin b drawing up a table of values: Using these values the graph can be drawn, as shown: 10

7 = Eample 3 (a) Draw the curve with equation = +. Describe how the curve is related to the curve with equation =. (a) A table of values has been completed: The graph is shown below: 105

8 13. = This curve is a translation of the curve = along the vector 0. Eercises 1. (a) Draw the graph with equation = + 1. State the gradient of this line.. Draw the line that has gradient 1 and -intercept (a) Draw the lines with equations = + 3 and = + 3. Describe the transformation that would map one line onto the other.. (a) Draw the curves with equations = and = 1. Describe how the two curves are related. 106

9 5. (a) Draw the curves with equations = + 3 and = 1. Describe the transformation that would map the first curve onto the second. 6. Without drawing an graphs, describe the relationship between the curves with equations, = + 1, = 5 and = (a) Cop and complete the following table: Draw the graph = +. (c) Describe how this curve is related to the curve with equation =. 8. (a) Draw the curve =. (c) On the same diagram, also draw the curves with equations = 1 and = +. Describe how the three curves are related. 9. (a) Draw the graphs with equations = + and = Describe the transformation that would map one curve onto the other (a) Draw the curves with equations = and = Describe the relationship between the two curves. 11. (a) The diagram shows the graph with equation =. = Cop the diagram and, on the same aes, sketch the graph with equation =. 107

10 13. Curve A is the reflection in the -ais of =. = A What is the equation of curve A? (c) Curve B is the translation, one unit up the -ais, of =. B (0, 1) What is the equation of curve B? (d) The shaded region is bounded b the curve = and the line =. = = Write down two of the inequalities below which together full describe the shaded region. < < 0 < < 0 > > 0 > > 0 (KS3/98/Ma/Tier 6-8/P1) 108

11 13.3 Graphs of Cubic and Reciprocal Functions In this section we look at the graphs of cubic functions, i.e. the graphs of functions whose polnomial equations contain 3 and no higher powers of. We also look at the graphs of reciprocal functions, for eample, Eample 1 (a) Draw the graph of =. Describe the smmetr of the curve. = 1, = 3 and =. (a) First complete a table of values: = The graph is shown below: = The graph has rotational smmetr of order about the point with coordinates (0, 0). 109

12 13.3 Eample Draw the graph with equation = 3 3. Completing a table of values gives: = The graph is shown below: = Eample 3 (a) Draw the curve with equation = 8. On the same diagram, draw the line with equation = +. (c) Write down the coordinates of the points where the line crosses the curve. 110

13 (a) Completing a table of values gives: = Note that 8 is not defined when = 0. These values can then be used to draw the graph below. 16 = = =

14 13.3 The line = +, which passes through ( 6, ), (0, ) and (8, 10). has been added to the graph above. (c) The coordinates of these points are (, ) and (, ). Eercises 1. (a) On the same set of aes, draw the graphs with equations, = 3 + 5, = 3 1 and = 3. Describe how the graphs are related.. (a) Draw the graph of the curve with equation = 3. Describe the smmetr of the curve. ( ) 3. (a) Draw the graph of the curve with equation = Describe how the curve relates to the graph of = 3. (c) Describe the smmetr of the curve = ( + 1) 3. ( ). (a) Draw the graph of the curve with equation = 3. Describe the smmetr of this curve. 5. (a) Draw the graphs of the curves with equations = = 3. 3 and Describe the transformation that would map one curve onto the other. 6. (a) Cop and complete the following table: (c) Use these values to draw the graph of the curve with equation = 1. Describe the smmetr of this curve. 11

15 7. On the same set of aes, draw the curves with equations, = 1, = and =. 8. (a) On the same set of aes, draw the curve with equation = 6 and the line with equation = 7, for values of from 1 to 7. Write down the coordinates of the points where the line intersects the curve. 9. Determine, b drawing a graph, the coordinates of the points where the line with equation = 3 intersects the curve with equation = 10. Use values of from to Determine, graphicall, the coordinates of the points where the curve intersects the curves with equations, (a) =, = 3. = Solving Non-Linear Equations In this section we consider how to solve equations b using graphs, trial and improvement or a combination of both. Eample 1 Solve the equation 3 + = 6 b using a graph. The graph = 3 + should be drawn first, as shown. A line can then be drawn on the graph from 6 on the -ais, across to the curve and down to the -ais. This gives an approimate solution between 1.6 and 1.7, so graphicall we might estimate to be

16 13. =

17 13. Eample Determine a solution to the equation 3 + = 6 correct to decimal places. The previous eample suggested graphicall that there is a solution of the equation between = 16. and = 17.. We will now use a trial and improvement method to find to decimal places, using = 16. as a starting value in the process. Trial 3 + Comment < < < < < < < < < < < At this stage we can conclude that the solution is = 163. correct to decimal places. Eample 3 Use a graph and trial and improvement 3 to solve the equation + = 10. The graph indicates that there will be a solution that is a little less than, approimatel = A trial and improvement approach is now used to determine to a greater degree of accurac

18 13. 3 Trial + Comment < < < < < < < < < < < < < 1.87 The solution is = 187. correct to decimal places. 3 Note: The equation + = 10 had just one solution. However, in general, there ma be more than one solution. For eample, the diagram shows that the equation 3 + = 01. has three solutions. 1.5 = = Eercises 1. Use a graph to determine the solutions to the equation + = 6.. (a) Draw the graph =. Use the graph to determine approimate solutions to the equations: (i) = 8, (ii) = The following graph is for = 3 6. Use the graph to solve the following equations: 3 3 (a) 6 = 8 6 = (c) 6 = (d) 6 = 116

19 0 3 = The equation + = 1000 has a solution close to = 10. Use trial and improvement to obtain the solution correct to decimal places. 5. The equation + = 5 has a solution between = 3 and =. Find this solution correct to decimal places. 6. Use a graphical method followed b trial and improvement to determine both solutions of the equation + 6 = 8, correct to decimal places The equation + = 8 has solutions. (a) Use a graph to determine approimates values for these solutions. Determine these solutions correct to decimal places using trial and improvement. 8. The equation 8 3 = 5 has 3 solutions. Determine each solution correct to 1 decimal place. 117

20 The equation 1 + = 1 has 1 solution. Determine this solution correct to decimal places Determine each of the solutions of the equation = correct to decimal places. 11. The table below shows values of and for the equation = + 5. (a) Cop and complete the table The value of is 0 for a value of between 1 and. Find the value of, to 1 decimal place, that gives the value of closest to 0. You ma use trial and improvement, as shown (KS3/96/Ma/Tier 6-8/P1) 1. Enid wants to find the roots of the equation = The roots are the values of which make the equation correct. Enid works out values of and She also works out the difference between each pair of values b subtracting the value 10 5 from the value of. Enid notes whether this difference is positive or negative Difference Positive Positive Positive Negative Negative (a) One root of the equation = 10 5 lies between = 0 and = 1. Use the table to eplain wh. 118

21 Enid then tries 1 decimal place numbers for Difference Between which two 1 decimal place numbers does the root lie? (c) Tr some decimal place numbers for. Show all our trials in a table like the one below. Find the two values of between which the root lies. Write down all the digits ou get for the values of, 10 5 and Difference Difference (d) Between which two decimal place numbers does the root lie? (KS3/95/Ma/Levels 5-7/P1) 119

22 = The graph shows = +. (a) Solve the equation + = 0 using the graph. Give our answers to decimal places. Give an eample of another equation ou could solve in a similar wa using the graph. (c) The equation = 0 cannot be solved using the graph. Wh not? Kell used an iterative method to find a more accurate solution to the equation + = 0. Kell's method was n + 1 = n + 10

23 (d) Eplain how Kell's method relates to the equation + = 0. Kell started with 1 = 1 used her iterative method four times. She got these results (e) Steve used a different iterative method. Steve's method was n + 1 = He started with 1 = 1. Work out, 3, and 5 and write them showing all the digits on our calculator. n n. (KS3/95/Ma/Levels 9-10) 13.5 Quadratic Inequalities In the first section of this unit we considered linear inequalities. In this section we will consider quadratic inequalities and make use of both graphs and the factorisation that ou used in Unit 11. We begin with a graphical approach. Eample 1 (a) Draw the graph = Use the graph to solve the inequalit The graph is shown below. Note that The graph shows that when 5 or = 0 at = 5 and =. 11

24 = Eample Solve the inequalit 6< 0 Factorising gives: 6 = ( 6) = 6 So 6 = 0 when = 0 or = 6. Sketching the graph as shown indicates that the solution is 0 < < Eample 3 Solve the inequalit 5 > 0 1

25 Factorising gives: ( )( + ) 5 = 5 5 So 5 = 0 when = 5 or = 5. 5 = 5 The sketch shows that 5 > 0 will be satisfied when 5< < Eercises 1. (a) Draw the graph = + 3. Solve the inequalit (c) Solve the inequalit + 3< 0.. Use a graph to solve the inequalit 1 < Use a graph to solve the inequalit (a) Factorise 5. Sketch the graph of = 5. (c) State the solution of the inequalit 5 < Solve the following inequalities: (a) + 5 > (c) < 0 (d) 0 6. (a) Factorise 9. Sketch the graph of = 9. (c) State the solution of the inequalit 9 > 0. 13

26 Solve the inequalities: (a) 36 < (c) 16 0 (d) 81 < 0 8. (a) Factorise 5 1. Sketch the graph of = 5 1. (c) State the solution of the inequalit Solve the inequalities: (a) 6 7 < (c) > 0 (d) (a) Factorise Sketch the graph of = (c) State the solution of the inequalit < 0. ( ) 11. Denise and Luke are using the epression nn+ 1 numbers. For eample, the triangular number for n = is + 1 to be 10. nn+ 1 (a) Denise wants to solve the inequalit 300 < two triangular numbers between 300 and 360. What are these two triangular numbers? You ma use trial and improvement. to generate triangular ( ), which works out ( ) < 360 to find the Luke wants to find the two smallest triangular numbers which fit the ( ) > inequalit nn What are these two triangular numbers? You ma use trial and improvement. (KS3/95/Ma/Tier 6-8/P1) 1

27 13.6 Equations of Perpendicular Lines In this section we consider the relationship between perpendicular lines. If one line has gradient m, m 0, a line that is perpendicular to it will have gradient 1 m. Note: The eamples that follow make use of the general equation = m + c for a straight line with gradient m and -intercept c. Eample 1 A line passes through the origin and is perpendicular to the line with equation = 7. Determine the equation of the line. The line's equation = 7 can be rewritten in the form = +7 showing that it has gradient 1. A perpendicular line will have gradient 1 1 = As it passes through the origin, we know = 0 when = 0. Substituting these values into the equation gives, 0 = 0 + c so c = 0 Hence the equation is Eample =. 1 and so its equation will be = + c. A line passes through the points with coordinates (, 6) and (5, 1). A second line passes through the points with coordinates (0, 3) and (7, 6). Are the two lines perpendicular? Gradient of first line = ( 1 ) 6 5 = 7 3 Gradient of second line = =

28 13.6 But as 3 7 = 1, the lines are perpendicular. 7 3 Note: This eample illustrates an alternative wa of looking at the relationship between the gradients of perpendicular lines, namel that the product of their gradients is 1. For eample, 7 3 = Eercises 1. (a) Draw the line with equation = 1. Determine the equation of a perpendicular line that passes through the origin. (c) Draw this line and check that it is perpendicular.. The equations of 5 lines are given below. A = 8 5 B = 3 + C D E = = 3 = Which line or lines are: (a) perpendicular to A, perpendicular to B? 3. The points A, B, C and D have coordinates (1, 3), (6, 1), (3, 1) and (5, 6) respectivel. Show that AB is perpendicular to CD.. A quadrilateral has corners at the points with coordinates Q (7, 3), R (6, 5), S (, 3) and T (3, 1). Show that QRST is a rectangle. 5. The line with equation = 7 and a perpendicular line intersect at the point with coordinates (, 3). Determine the equation of the perpendicular line. 16

29 6. Are the lines with equations = and = perpendicular? 7. A line passes through the origin and the point (, 7). Determine the equations of the perpendicular lines that pass through: (a) the origin, the point (, 7) A line is drawn perpendicular to the line = + so that it passes through the point with coordinates (3, 3). (a) Determine the equation of the perpendicular line. Determine the coordinates of the point where the two lines intersect. 9. Two perpendicular lines intersect at the point with coordinates (, 6). One line has gradient. Determine the equations of the two lines 10. Two perpendicular lines intersect at the point with coordinates (6, 5). One line passes through the origin. Where does the other line intersect the -ais? 17

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