STRAND G: Relations, Functions and Graphs

UNIT G Using Graphs to Solve Equations: Tet STRAND G: Relations, Functions and Graphs G Using Graphs to Solve Equations Tet Contents * * Section G. Solution of Simultaneous Equations b Graphs G. Graphs of Common Functions G. Graphical Solutions of Equations G. Tangents to Curves CIMT, Plmouth Universit

UNIT G Using Graphs to Solve Equations: Tet G Using Graphs to Solve Equations G. Solution of Simultaneous Equations b Graphs Solutions to pairs of simultaneous equations can be found b plotting lines and finding the coordinates of the point where the intersect. The can be solved analticall but this introduces a more general method which we etend in later units. Worked Eample Solve the pair of simultaneous equations Solution + = 8 and + = First it can be helpful to write the two equations in the form =.... For the first equation, + = 8 = 8 (subtracting ) For the second equation, + = = (subtracting ) = 7 (dividing b ) Now two pairs of coordinates can be found for each line. For = 8 If =, = 8 so (, 8) lies on the line If = 8, = 8 8 For = 7 If =, = 7 = so (8, ) lies on the line. = 7 so (, 7) lies on the line. CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet If =, = 7 = 7 = so (, ) lies on the line. These points are then used to plot the lines shown below. 8 7 = 8 5 = 7 5 7 8 9 The two lines intersect at the point (, 5), so the solution is = and = 5 Note We can easil solve these equations analticall b writing + = 8 + = + = subtract from to give = 5 Substituting back in either equation gives ; for eample, Hence solution at (, 5). = 8 = 8 5 = Eercises. Use the graph below to solve the simultaneous equations. (a) = 8 = 8 = = (c) = (d) = = = 7 (e) = (f) = 7 = = CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet = 8 = 7 9 8 7 5 = = 9 8 7 5 5 7 8 9 5 7 8 =. (a) Draw a set of aes with -values from to and -values from 5 to 5. Write down the coordinates of three points on the line = and use them to draw the line =. (c) On the same set of aes draw the lines = 5 and = +. (d) Write down the solution of each set of simultaneous equations. (i) = = 5 (ii) = = + (iii) = 5 = +. (a) Write the equations = and + = in the form =... (c) Draw the graphs of both equations. What is the solution of the simultaneous equations? CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet. A discount store sells CDs and DVDs. The price of ever CD is and the price of ever DVD is. (a) Jane bus CDs and DVDs which cost a total of. Write down an equation involving and using this information. Write down an equation in the form =.... (c) Christopher bus CDs and DVDs which cost a total of. Write down a second equation using this information. (d) Write this equation in the form =.... (e) (f) (g) Draw the graphs of both equations on the same set of aes. What is the price of a CD? What is the price of a DVD? 5. An enterprising schoolbo charges to wash a car and 5 to wash and polish a car. One da he earns 8 and cleans 8 cars. Let be the number of cars that he onl washed, and the number of cars washed and polished. (a) Write down two equations involving and. Write these equations in the form =.... (c) Draw the graphs of both equations and find out how man cars were washed and polished.. 7 5 = + + = 7 5 The diagram shows the graphs of the equations = + and + = 7 Use the diagram to solve the simultaneous equations = + + = 7 CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet 7. A longlife batter and a standard batter were both tested for their length of life. The longlife batter lasted for hours. The standard life batter lasted for hours. (a) The combined length of life of the two batteries was hours. Eplain wh + = The longlife batter lasted hours longer than the standard batter. Write down another equation connecting and. The graph of + = has been drawn below. + = 8 8 (c) Complete a cop of the table of values for our equation in and use it to draw the graph of our equation. 7 (d) Use our graphs to find the length of life of each tpe of batter. CIMT, Plmouth Universit 5

G. UNIT G Using Graphs to Solve Equations: Tet 8. For off-peak electricit, customers can choose to pa b Method A or Method B. Method A: Method B: 8 per quarter plus p per unit. per quarter plus p per unit. A customer uses units. The quarterl cost is. The cost,, b Method B is =. +. (a) (i) Complete the table of values of =. +. 5 (ii) Draw the graph of =. +. Method A Cost ( ) Cost ( ) Number of units used For a certain number of units both methods give the same cost. Use the graphs to find (i) (ii) this number of units, this cost. (c) Cop and complete the following statement "Method... is alwas cheaper for customers who use more than... units." CIMT, Plmouth Universit

G. 9. (a) UNIT G Using Graphs to Solve Equations: Tet 8 7 5 5 (i) The graph of 5 + = is shown on the diagram above. On a cop of the diagram, draw the graph of =. (ii) Use the graphs to find the solution of the simultaneous equations 5 + = = Give the value of and the value of to one decimal place. Calculate the eact solution of the simultaneous equations 5 + = =. The line with equation + = is drawn on the following grid. Solve the simultaneous equations b a graphical method. = + = CIMT, Plmouth Universit 7

G. UNIT G Using Graphs to Solve Equations: Tet 5 5 7 G. Graphs of Common Functions Linear Functions Linear functions are alwas straight lines and have equations which can be put in the form = m + c c Gradient = m Quadratic Functions Quadratic functions contain an term as well as multiples of and a constant. Some eamples are: = = + 5 = The following graphs show eamples. 8-8 - 8 - = = 8 = + + CIMT, Plmouth Universit 8

G. UNIT G Using Graphs to Solve Equations: Tet Note that each curve has either a maimum or a minimum point which lies on its ais of smmetr. The curve has a maimum point when the coefficient of is negative as in the second eample, or minimum if the coefficient of is positive. Also the curve can intersect the -ais twice, just touch it once or never meet the -ais. Cubic Functions Cubic functions involve an term and possibl, and constant terms as well. Some eamples are: =, = + + 8, = 5, = + The graphs below show some eamples. (a) (c) = = + = + (d) (e) (f) = + = = + + The graph of a cubic function can intersect the -ais once as in eamples (a), and (e), touch the ais once and intersect it once as in eample (c) or intersect the -ais three times as in eamples (d) and (f). In eamples (c), (d) and (f) the curve has a local minimum and a local maimum. Note how the shape of the curve changes when a (a) and (e). is introduced. Compare eamples CIMT, Plmouth Universit 9

G. UNIT G Using Graphs to Solve Equations: Tet Reciprocal Functions Reciprocal functions have the form of a fraction with as the denominator. Eamples of reciprocal functions are: = =, =, = 5 The graphs below show some eamples. = = = The curves are split into two distinct parts. The curves get closer and closer to the aes as is clear in the diagrams. The curves have two lines of smmetr, = and =. Eercises. State whether each equation below would produce the graph of a linear, quadratic, cubic or reciprocal function. (a) = = (c) = (d) = + + (e) = + (f) = 7. Each of the following graphs is produced b a linear, quadratic, cubic or reciprocal function. State which it is for each graph. (a) (c) (d) (e) (f) CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet. One of the graphs shown below is = +. Which one? A B C D. Which of the graphs shown below are reciprocal functions? A B C D 5. Each equation below has been plotted. Select the correct graph for each equation. (a) = + A = (c) = B (d) = + C D CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet. Match each graph below to the appropriate equation. A = + C B = + = D = + (a) (c) (d) 7. (a) Which of the following equations are illustrated b the graphs shown? Write the equation illustrated beside the number of each graph. = = = = = + = (i) (ii) (iii) (iv) Sketch a graph of the equation = + on a cop of this graph CIMT, Plmouth Universit

UNIT G Using Graphs to Solve Equations: Tet G. Graphical Solutions of Equations Equations of the form f( ) = g( ) can be solved graphicall b plotting the graphs of = f( ) and = g( ). The solution is then given b the -coordinate of the point where the intersect. Worked Eample Find an positive solutions of the equation b a graphical method. Solution = + Completing the table below provides the points needed to draw the graphs = +. + Infinit.5.5.5..5.5.7..5.5.5.9 Where necessar the values have been rounded to decimal places. The graph below shows = and = +.. 9. The curves intersect where = 5. and so this is the solution of the equation. = and = 5 = +.5. CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet Worked Eample The graph below represents the function ( ) = f f() 8 f() = - - - - 5 - - - Use the graph to determine (a) the value of f( ) when = the value of f( ) when =. 5 (c) (d) (e) the value of for which f( )= the minimum value of f( ) the value of at which f( ) is a minimum (f) the solution of = 5 (g) the interval on the domain for which f( ) is less than. CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet Solution Using the graph: (a) f ( ) = 5 ( ) f 5.. (c) Intercepts with -ais are = 8., 8. (d) f min = 5. (e) = 5. (f) =. 7 and.7 (g) < < Note The domain of a function is the values of for which the function is defined. This is covered in Unit G. Worked Eample Given that = 9 + (a) cop and complete the table below (c) using a scale of cm to represent unit on the -ais and cm to represent 5 units on the -ais, draw the graph of = 9 + for use our graph to solve the equation 9 + = 5 Solution (a) Missing values: f ( ) f ( ) ( ) + = = 9 5 ( ) = ( ) 9 + = 5 CIMT, Plmouth Universit 5

G. UNIT G Using Graphs to Solve Equations: Tet 5 5 5 - - 5-5 (c) Using the intersection of = 5 with = 9 + solution of gives estimates of the 9 + = 5 as = and = 55. (see graph above). Worked Eample (a) The grid on the following page shows the line, l, which passes through the points Q (, ) and R (, ). (i) Determine the gradient of the line, l. (ii) Write down the equation of the line, l. ( ) = + The table below shows three of the values of f for values of from to. CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet (i) Cop the table and insert the missing values of f( ). ( ) = + (ii) On a cop of the grid below, draw the graph of f. (iii) Using the graphs, write down the coordinates of the points of intersection of the line, l, and the graph of f ( ). f() R l - 5 Q - - - Solution (a) (i) Gradient = ( ) = = (ii) -intercept is, so equation of l is = + (Alternative method: equation is of the form = m + c = + c; to pass through the point (, ), = + c c = = ) CIMT, Plmouth Universit 7

G. UNIT G Using Graphs to Solve Equations: Tet (i) (ii) f( ) f() R l - 5 Q - - - (iii) Points of intersection at (, ) and (, ). Eercises. Draw the graph of = for. Use the graph to solve the equations = and 5 =.. Solve the quadratic equation = b plotting the graphs and = +.. Find the -coordinates of the two points where the lines = and = + intersect. Write down the quadratic equation which has the two solutions ou found from the points of intersection. = CIMT, Plmouth Universit 8

G. UNIT G Using Graphs to Solve Equations: Tet. Find the solutions of the following equations (a) = = (c) = 8 (d) = 8 5. Describe different was to find solutions of the equation 8 + 5 =. Draw the graph = + for. Use the graph to solve the following equation, + a = if (a) a = a = (c) a = For what value of a is there onl one solution? For what value of a are there no real solutions to the equation? 7. Use a graphical method to solve the equation You must show all our working. = CIMT, Plmouth Universit 9

G. 8. UNIT G Using Graphs to Solve Equations: Tet 8 The diagram above shows the graph of = +. (a) Use the graph to find the solutions of the equation + = B drawing the graph of = on a cop of the diagram, find the solution of the equation = (c) Use the graph to find solutions of the equation + = 9. 7 The table above shows some values of = for values of from to. (a) What are the missing values of? On a graph, plot the points recorded in our completed table at (a) above, and draw a smooth curve through the points. (c) Use our graph to find the values of for which =. CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet. - - - - - The diagram above shows the graph of the function = p + q + r. (a) Determine the values of p, q and r. State TWO was in which the graphs of the functions = p and = p + q + r are similar. (c) State ONE wa in which the graphs of the two functions is different.. (a) Given that f( ) = +, cop and complete the table below. f( ) Using cm to represent unit on both aes, draw the graph of ( ) = + f for. ( ) = + (c) On the graph of f the values from the table shown below. g ( ), draw the graph of g ( ) = using (d) Using the graphs, write down the coordinates of the points where the two graphs intersect. CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet ( ) =. The diagram below shows the graph of the function f for a b. The tangent to the graph at (, ) is also drawn. Use the graph to determine the (a) values of a and b which define the domain of the graph. values of for which = (c) coordinates of the minimum point on the graph (d) whole number values of for which < ( ) = (e) gradient of f at =. f() 5 - - - - - -. (a) Given that =, cop and complete the table below. 5..5 (c) Using scales of cm to represent unit on the -ais, and cm to represent unit on the -ais, draw the graph of the function for Using the graph (i) solve the equation, = (ii) determine the values of for which. CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet (d) Using the same aes and scales, (i) draw the graph of = (ii) write down the equation in whose root is given b the intersection of the graphs, = and =. (CXC) G. Tangents to Curves A tangent is a line that touches a curve at one point onl, as shown opposite. The gradient of the tangent gives the gradient of the curve at that point. The gradient of the curve gives the rate at which a quantit is changing. For eample, the gradient of a distance-time curve gives the rate of change of distance with respect to time, which gives the velocit. CURVE TANGENT Worked Eample Draw the graph of = for. Draw tangents to the curve at = and =. Find the gradients of these tangents. Solution The graph of = is shown below. The tangents have been drawn at = and =. 8 = Tangent at = Tangent Tangent at = 8 Using the triangles shown under each tangent, show that the gradients of both tangents are. CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet Worked Eample The height, h, of a ball thrown straight up in the air varies so that at time, t, h = 8t 5t. Plot a graph of h against t and use it to find: (a) the speed of the ball when t =., the greatest speed of the ball. Solution The table below gives the values needed to plot the graph. The graph is shown below. t (s)....8.... h (m)....... h..8 Tangent at t =. Tangent at t =. (a) A tangent has been drawn at the point where t =.. The gradient of 8. this tangent is =.. So the speed of the ball is m/s....8... t The speed of the ball is a maimum when the curve is steepest, that is at t = and t =.. At t =. the gradient is. = 8. So the. speed is 8 m/s. The ' ' sign indicates that the ball is moving down rather than up. You can sa the ball moves down with speed 8 m/s or that the velocit of the ball is 8 m/s. Worked Eample The following graph represents the cooling curve for a certain liquid. Use the graph to estimate (a) the temperature when the time, t, is secs. the gradient of the curve when the time, t, is secs. CIMT, Plmouth Universit

G. UNIT G Using Graphs to Solve Equations: Tet T Temperature ( o C) 8 - - - 8 8 8 Time (sec) t Solution (a) From the graph, temperature C at time secs. From the graph (see net page), gradient = 5C per sec. CIMT, Plmouth Universit 5

G. UNIT G Using Graphs to Solve Equations: Tet T Temperature ( o C) 8 Gradient at t = - - - 8 8 8 Time (sec) t Worked Eample The graph shows how the velocit of a car changes. Find: (a) the time when the acceleration of the car is zero, the acceleration when t =. Solution The acceleration of the car is given b the gradient of the velocit-time graph. There are points where the gradient is zero, at t =, t =, t = 5 and t =. At each of these points a horizontal tangent can be drawn to the curve as shown opposite. Velocit (m/s) Speed (m/s) Time (s) Time (s) A tangent has been drawn to the curve at t =. The gradient of this curve is CIMT, Plmouth Universit =., so the acceleration is. m/s.

G. UNIT G Using Graphs to Solve Equations: Tet Eercises. (a) Draw the graph of = for. B drawing tangents find the gradient of the curve at =, =, =, = and =. (c) Comment on an patterns that are present in our answers.. The height, h, of a ball at time, t, is given b, h = t 5t. The ball travels straight up and down. (a) Draw a graph of h against t for t. Use the graph to find the velocit of the ball when t =.5 and t =.. (c) Find the maimum speed of the ball.. The graph below shows how the temperature of a can of drink increases after it has been taken out of a fridge. T ( C) 5 5 5 Time (mins) Find the rate of change of temperature with respect to time, when; (a) t =, t =, (c) t = 5.. A car moves so that its velocit, v, and time, t, is given b v = t 8t +. (a) Plot a graph of velocit against time for t. Find the gradient of the curve when t =,,, and. (c) Use our results to to sketch a graph of acceleration against time for the graph. CIMT, Plmouth Universit 7

G. UNIT G Using Graphs to Solve Equations: Tet 5. Displacement 5 7 Time (s) The graph shows how the displacement of an object varies with time. (a) Cop the graph and b drawing tangents estimate the velocit of the object when t =,,,,, 5, and 7. Use our results to part (a) to draw a velocit-time graph. (c) Consider our velocit-time graph and sketch an acceleration-time graph.. Draw the graph of = for. (a) Find the gradient of the curve at =,,,,, and. Can ou predict how to calculate the gradient of = for an value of? 7. Draw a graph of = sin for. (a) For what values of is the gradient of the curve zero? Find the maimum and minimum values of the gradient of the curve = sin, and state the values of for which the are obtained. (c) Use these results to draw a graph of the gradient of = sin against. 8. For each of the following velocit-time graphs, sketch an acceleration-time graph showing the maimum and minimum values of the acceleration. (a) v v 5 5 5 t 5 t CIMT, Plmouth Universit 8

G. UNIT G Using Graphs to Solve Equations: Tet (c) (d) v v 5 t 5 t 9. Here is a velocit-time graph of a car travelling between two sets of traffic lights. Velocit m s ( ) 8 Calculate an estimate for the acceleration of the car when the time is equal to seconds.. The temperature, K, of a liquid t minutes after heating is given in the table below. t (time in minutes) K (Temp. in C) 5 8 9 7 5 (a) (i) Using a scale of cm to represent minutes on the horizontal ais and a scale of cm to represent degrees on the vertical ais, construct a temperature-time graph to show how the liquid cools in the minute interval. Draw a smooth curve through all the plotted points. Use our graph to estimate (i) the temperature of the liquid after 5 minutes (ii) the rate of cooling of the liquid at t = minutes. CIMT, Plmouth Universit 9