74 Maths Quest 10 for Victoria

Transcription

1 Linear graphs Maria is working in the kitchen making some high energ biscuits using peanuts and chocolate chips. She wants to make less than g of biscuits but wants the biscuits to contain at least 8 g of carbohdrates. The peanuts provide % of their mass in carbohdrates, while the chocolate chips provide 6% of their mass in carbohdrates. Maria wants to find all the combinations of peanuts and chocolate chips that can be used to meet these requirements. In this chapter, we will revise linear equations and ou will be able to model the carbohdrate proportions to provide possible answers to Maria s problem.

2 7 Maths Quest for Victoria Plotting linear graphs You will recall that a linear graph is drawn on a Cartesian plane, with two aes ( and ), meeting at the origin (, ). The aes divide the plane into four regions, or quadrants. A location (point) is specified b its - and -coordinates, such as (, ). This information is summarised in the figure below. Quadrant Quadrant Quadrant (, ) Quadrant Normall, we draw a linear graph from a given rule, such as = +. This line is also shown in the figure. Given a rule or equation, we can create a table of values. Each pair of coordinates is then plotted and the points connected to form a straight line. WORKED Eample Plot the linear graph defined b the rule = for the range of -values, 8,... 8,. THINK Create a table of values using the given -values. WRITE Find the corresponding -values b substituting each -value into the rule. If =, = ( ) = Plot the points on a Cartesian plane and rule a straight line through the points. Etend the line slightl in both directions to show that it has an infinite length. =

3 remember remember Chapter Linear graphs 7. The Cartesian plane is a grid, consisting of two aes ( and ), meeting at the origin (, ).. A location (point) is specified b its - and -coordinates.. A linear graph consists of an infinite set of points that can be joined to form a straight line.. To plot a linear graph, the coordinates of onl two points are needed, although several points are often plotted as a check.. A linear rule or equation can be used to obtain the coordinates of points that belong to its graph. WORKED Eample A Plotting linear graphs Plot the linear graphs defined b the following rules for the given range of -values. Rule -values = +, 8,... 8, = +, 8,... 8, = +, 8,... 8, = +, 8,... 8, = +,,..., 6 =,,..., 7 =. +, 8,... 8, 8 =, 8,... 8, 9 = +, 8,... 8, = 7, 8,... 8, SkillSHEET Plotting linear graphs EXCEL Plotting from a table DIY EXCEL Plotting from a rule DIY. Mathcad Spreadsheet Spreadsheet MATHS MATHS QUEST C H A L LL E N G G E E What are the net three numbers in the fifth column of this number pattern? a Divide the eight numbers listed below into two groups of four where the difference between the sums of the numbers in each group is as small as possible. 7,, 67,,,, 6, 9 b How can ou tell that these numbers cannot be divided into two groups of four where the sums are eactl equal?

4 76 Maths Quest for Victoria Sketching linear graphs The rule or equation for a linear graph can be epressed in the form = m + c, where m represents the gradient (slope) of the straight line and c represents the -intercept. This information is useful because the sign of the gradient indicates whether the line slopes upward (positive gradient) or downward (negative gradient) as we move in the positive direction. In addition, we can immediatel see where to mark the -intercept on the coordinate aes. For eample, the graph of the line with equation = + has a gradient m = and -intercept c =. WORKED Eample Find the gradient and -intercept of the following straight lines. a = 8 b + = = + THINK WRITE a Write the rule. a = 8 Compare it to = m + c. Gradient: m = -intercept: c = 8 b Write the rule. b + = Rearrange the equation so it can be = compared to = m + c. Subtract = -- from both sides then divide both sides b. = + -- Gradient: m = -intercept: c = -- Note that a linear equation can be epressed in different forms. The following represent the same equation. General form: = + Factor form: ( ) = ( + ) Other forms: + = = Sketching using the intercept method The intercept method lets us sketch the graph of an linear equation b finding two particular points, the -intercept and the -intercept. To find the -intercept, we know that the -coordinate of this point will alwas be zero, so we can substitute = into the equation to find. Likewise, to find the -intercept we substitute = into the equation. If the equation is in the form = m + c, we can save ourselves some work b looking at the equation to find the -intercept, c.

5 Chapter Linear graphs 77 WORKED Eample Sketch graphs of the following linear equations b finding the - and -intercepts. a + = 6 b = THINK WRITE a Write the equation. a + = 6 Find the -intercept b substituting =. Find the -intercept b substituting =. Rule a straight line passing through both points. -intercept: when =, + = 6 = 6 = -intercept is (, ). -intercept: when = () + = 6 = 6 -intercept is (, 6). 6 + = 6 b Write the equation. b = Find the -intercept b substituting =. Find the -intercept. The equation is in the form = m + c, so compare this with our equation to find the -intercept, c. Rule a straight line passing through both points. -intercept: when =, = = = -intercept is (, ). c = -intercept is (, ). =

6 78 Maths Quest for Victoria Sketching linear graphs of the form = m Linear equations in the form = m have the constant term, c, equal to. This means that the -intercept is at (, ) or the origin. Since the -intercept will also be at the origin, this provides us with onl one point to plot. To find a second point, we can choose an -value, sa, and find the corresponding -value. WORKED Eample Sketch the graph of =. THINK Write the equation. Find the - and -intercepts. Note: B recognising the form of this linear equation, ou can simpl state that the line passes through the origin, (, ). Find another point to plot b finding the -value when =. Plot the two points (, ) and (, ) and rule a straight line through them. WRITE = -intercept: when =, = = -intercept: when =, = Both the - and -intercepts are at (, ). When =, = = Another point is (, ). = Lines parallel to the aes Sketching the linear graphs of the form = c and = a The equations that we have looked at so far contained both an -term and a -term. It is possible to have an equation for a straight line that contains onl an -term, or onl a -term. These equations can be written in the form = c or = a, where c and a are both constants. For eample, the equation of the straight line given b = does not contain an -term. (It is of the form = m + c, where m =.) This means that its graph does not have an -intercept. The onl wa this is possible is if the graph is parallel to the -ais. We are also saved some work because the -intercept is given b the equation itself! That is, =. =

7 Chapter Linear graphs 79 Similarl the graph of the equation = does not have a -intercept and is parallel to the -ais. = Graphs of equations in the form = c have a gradient of zero and are parallel to the -ais. Graphs of equations in the form = a have an undefined (infinite) gradient and are parallel to the -ais. WORKED Eample Sketch graphs of the following linear equations. a = b = THINK WRITE a Write the equation. a = The -intercept is. As does not appear in the equation, the line is parallel to the -ais. -intercept = Sketch a horizontal line through (, ). = b Write the equation. b = The -intercept is. As does not appear in the equation, the line is parallel to the -ais. -intercept = Sketch a vertical line through (, ). =

8 8 Maths Quest for Victoria Graphics Calculator tip! Graphing a linear equation A linear graph can be displaed on a graphics calculator. To do this ou should have the equation in the form = m + c. Suppose we were to sketch the graph from worked eample (a) + = 6.. First write the equation as = On our graphics calculator press Y= and enter this function. (Use X, T, θ, n or ALPHA [X] to enter the variable,.). Press GRAPH and the graph should be displaed. You ma need to adjust the window settings on our calculator to see the graph properl.. The TRACE function can be used to eplore the line.. To locate the -intercept, press nd [CALC], select : value and enter (for = ). 6. To locate the -intercept, press nd [CALC] and select : zero. For the prompt Left bound? move to the left of the -intercept using the arrow kes. Press ENTER. For the prompt Right bound? move to the right of the -intercept. Again press ENTER. Finall, at the prompt Guess? move close to the -intercept and press ENTER to displa the zero or coordinates for the -intercept.

9 remember remember Chapter Linear graphs 8. A linear equation can be epressed in the form = m + c, where m represents the gradient (slope) of the straight line and c represents the -intercept.. A straight line with a positive gradient slopes upward as we move in the positive direction. A straight line with a negative gradient slopes downward as we move in the positive direction.. Linear equations ma be written in several different forms.. The intercept method allows us to sketch the graph of an linear equation b finding two specific points, the -intercept and -intercept.. Graphs of equations in the form = m pass through the origin. To find the second point, substitute a chosen -value into the equation to find the corresponding -value. 6. Graphs of equations in the form = c have a gradient of zero and are parallel to the -ais. 7. Graphs of equations in the form = a have an undefined (infinite) gradient and are parallel to the -ais. B Sketching linear graphs WORKED Eample WORKED Eample WORKED Eample WORKED Eample Find the gradient and -intercept of the following straight lines. a = + b = 7 c = + d = 8 e = f = g =.6 +. h = 8 i = 7 Sketch graphs of the following linear equations b finding the - and -intercepts. a = b + = c + = d = e 8 = f + = g + 6 = h + 8 = i + = j + = k 9 + = 6 l 6 = Sketch the graphs of the following linear equations. a = b = c = d = -- e = -- f = -- Sketch the graphs of the following linear equations. a = b = c = d = e = f = g = h = Use technolog to graph each of the following. State the -intercept and -intercept for each. a ( + ) = ( + ) b ( ) = ( ) c ( + ) = ( + ) d ( ) = ( ) e ( + ) = ( + ) f ( ) = ( + ) g ( + ) = ( ) h 8( ) = ( + ) i ( +.) = (.) j.( ) = 6.( ) -- SkillSHEET SkillSHEET Sketching linear graphs EXCEL Linear graphs Linear graphs.. Mathcad Spreadsheet GAMEtime

10 8 Maths Quest for Victoria Complete the table of values for the rule = + and use the range of -values given to sketch the linear graph. Complete the table of values for the rule = and use the range of -values given to plot the linear graph. 6 6 Find the gradient and the -intercept of the equation = 6 7. Find the gradient and -intercept of the equation = --. Sketch the graph of =. 6 Find the gradient of the linear graph with equation ( + ) = ( 7). 7 For the equation in question 6 find the - and -intercepts. 8 Sketch the graph of = 6. 9 Sketch the graph of =. Sketch the graph of = 8. Finding linear equations It is possible to find a linear equation without drawing a graph, if certain information about the graph is known. The gradient intercept method Because we know that the general formula for a linear equation is = m + c, we can find the equation easil if the values of m and c are given. WORKED Eample 6 Find the equation of the straight line with gradient of and -intercept of. THINK Write the known information. State the values of m and c. Substitute these values into = m + c to find the equation. WRITE Gradient =, -intercept = m =, c = = m + c =

11 Chapter Linear graphs 8 The point gradient method If we know the value of the gradient and the coordinates of an point that belong to the straight line, we can also find its equation. This is because we can find the value of the -intercept, c, b substituting these coordinates into = m + c. Find the equation of the straight line passing through the point (, ) with gradient of. THINK WRITE Write the known information. Gradient is, point (, ). State the values of m, and. m =, (, ) = (, ) Substitute these values into = m + c and solve to find c. = m + c = () + c = + c WORKED Eample 7 Substitute m = and c = 6 into = m + c to find the equation. 6 = c Equation is = 6. We can also use this method to find the equation of a line if we are given the coordinates of two points on the line. We must first find the gradient of the line joining the two points using the gradient formula rise m = or m = run WORKED Eample 8 Find the equation of the straight line passing through (, ) and (, ). THINK Write the general equation of a straight line. 6 7 Write the formula for calculating the gradient of a line between two points. Let (, ) and (, ) be the two points (, ) and (, ) respectivel. Substitute the values of the pronumerals into the formula to calculate the gradient. Substitute the value of the gradient into the general rule. Select either of the two points, sa (, ), and substitute its coordinates into = + c. Solve for c. State the equation b substituting the value of c into = + c. WRITE = m + c m = ( ) m = ( ) 6 = = = + c Point (, ): = + c = + c = c Equation is = +.

12 8 Maths Quest for Victoria Finding a linear equation from a graph We can use the graph to find a linear equation, again b finding the values of m and c. Since c is actuall the -intercept, we can read this directl from the graph. The gradient can be found b looking at the - and -intercepts because the form a triangle with the coordinate aes. Remember that the gradient of a straight line can be found b rise calculating run -intercept Run Rise -intercept It is necessar to look at the graph to decide if the gradient is positive or negative. A straight line with a positive gradient slopes upward as we move in the positive direction. A straight line with a negative gradient slopes downward as we move in the positive direction. Find the equation of the straight line shown in the graph. THINK WRITE State the -intercept, c. c = 6 WORKED Eample 9 Look at the triangle formed b the graph and the coordinate aes and state the size of the rise and run. Calculate the gradient, m. It will be negative since the line slopes downward as we move in the positive direction. Rise = 6 Run = rise m = run 6 m = m = 6 Substitute m =, and c = 6 into = m + c to find the equation. = m + c = + 6 Lines through the origin When the graph of a straight line passes through the origin, both the - and -intercepts equal zero. In order to find the gradient in cases like these, we require the coordinates of another point on the graph. This will allow us to form a right-angled triangle and we can calculate the gradient, m, as before.

13 Chapter Linear graphs 8 Find the equation of the straight line shown in the graph. THINK WRITE State the -intercept, c. c = Form a right-angled triangle between the given point and the coordinate aes, and state the size of the rise and run. Rise = Run = WORKED Eample Calculate the gradient, m. It will be positive since the line slopes upward as we move in the positive direction. Substitute m = -- and c = into = m + c to find the equation. remember remember rise m = run m = -- = m + c = -- + = -- (, ). The equation of a straight line can be found b substituting the values of the gradient (m), and -intercept (c), into = m + c.. If the -intercept is unknown, it can be found b substituting the - and -values of a given point into = m + c.. A graph can be used to find the equation of a straight line b calculating the rise gradient using m = and using the -intercept. run C Finding linear equations WORKED Eample 6 WORKED Eample 7 WORKED Eample 8 Find the linear equation given the information in each case below. a gradient =, -intercept = b gradient =, -intercept = c gradient =, -intercept = d gradient =, -intercept = e gradient =, -intercept = f gradient =., -intercept = g gradient =, -intercept =. h gradient = 6, -intercept = i gradient =., -intercept =. j gradient =., -intercept = 6. For each of the following find the equation of the straight line with the given gradient and passing through the given point. a gradient =, point = (, 6) b gradient =, point = (, 6) c gradient =, point = (, 7) d gradient =, point = (8, ) e gradient =, point = (, ) f gradient =, point = (, ) g gradient =, point = (, ) h gradient =, point = (,.) i gradient =., point = (6, 6) j gradient =., point = (, ) Find the equation of the straight line that passes through each pair of points. a (, ) and (, 6) b (, ) and (, ) c (, ) and (, ) d (, ) and (, ) e (, 6) and (, 6) f (, ) and (, 7) EXCEL Finding linear equations gradient and -intercept EXCEL Finding linear equations points SkillSHEET Spreadsheet Spreadsheet.

14 86 Maths Quest for Victoria WORKED Eample 9 Find the equations of the straight lines shown. a b SkillSHEET. EXCEL GC Spreadsheet Mathcad program c Finding d linear equations intercepts 8 Finding linear equations Gradient and finding linear equations e g 6 7 f h 6 EXCEL WORKED Eample Finding linear equations origin and another point Spreadsheet GAMEtime Linear graphs Find the equation of each of the straight lines shown. a b (, ) 6 (, 6) c d WorkSHEET. (, ) ( 8, 6) 8 6

15 Chapter Linear graphs 87 Who won the inaugural, 87 km, Sdne to Melbourne marathon in 98? gradient of and -intercept of A gradient of and passes through (, ) C The equations to the straight lines, in the form = m + c, that fit the given information and the letter beside each gives the puzzle answer code. passes through (, ) and (, ) m = and c = m = and passes through (, ) gradient of and -intercept of D E F passes through (, ) and (, ) m = and c = G H I passes through (, 7) and (, ) gradient of and passes through (, ) passes through (, 9) and (, ) L M N m = and c = gradient of and passes through (, ) passes through (, ) and (, ) O P R passes through (, 8) and has a gradient of passes through (, ) and (, ) gradient is and passes through (, 7) S T U m = and c = passes through (, ) with a gradient of passes through (, ) and (, ) V X Y = = = + = = + = = = + = + = + = + = = = + = + = + = = = + = = = + = + = = + = + = = = + = = + = + = + = = + = = + = = = + = = = + = + = + = + = = + = = + = + = = + = = + = + = = = = + = =

16 88 Maths Quest for Victoria Linear modelling In this section we will look at some practical situations using linear graphs and their equations. This will involve interpretation of worded problems, and the application of skills used in the previous eercises. WORKED Eample A buer for a department store can purchase table lamps in lots of. If she orders lamps the cost is $, while if she orders 6 lamps the cost is $8. Find: a A linear equation relating cost to the number of lamps ordered. b The cost of an order for lamps. THINK WRITE a Write the important points. a lamps cost $. 6 lamps cost $8. Define the pronumerals to be used. Note: An pronumerals can be used. Label a pair of coordinate aes with cost () on the vertical ais and choose a suitable scale. Plot the given information as the coordinate pairs (, ) and (6, 8 ). Rule a straight line through both points. Let represent the number of lamps ordered. Let represent the cost in dollars. Cost Use the two points to form a rightangled triangle and find the gradient. Use the point gradient method to find the equation b substituting m and one of the points into = m + c. Use the point (, ). rise m = run m = Number of lamps = = m + c = () + c = + c c = The equation is = or cost = number of lamps. b Write the equation just found. b = Substitute for the number of lamps (). = () = State the answer in the correct units. The cost is $.

17 Chapter Linear graphs 89 WORKED Eample A builder can complete a building in 6 das without an assistants, but if he hires five assistants he can build it in das. How man assistants would be required to complete the building in das? THINK Write the important information. Define the pronumerals to be used. Label a pair of coordinate aes with the number of das on the vertical ais and choose a suitable scale. Plot the information as the coordinate pairs (, 6) and (, ). Rule a straight line through both points. WRITE assistants take 6 das. assistants take das. Let represent the number of assistants. Let represent the number of das. Das Number of assistants rise Use the two points to form a rightangled triangle and find the gradient. m = run m = m = 8 Use the point gradient method to find the equation b substituting m and one of the points into = m + c. Use the point (, 6). = m + c 6 = 8() + c 6 = + c c = 6 The equation is = or das = 8 number of assistants + 6. Write the equation just found. = Substitute = and solve for. = = 8 6 = = 7 State the answer. He will require 7 assistants.

18 9 Maths Quest for Victoria remember. Linear modelling uses a linear equation to represent a situation. We can predict a -value from a given -value, or an -value from a given -value.. Alwas define each pronumeral used in the linear equation. D Linear modelling WORKED Eample WORKED Eample Springfield High School needs to purchase tetbooks for its ver large Year class. The supplier quotes a price of $ for books, and $ for books. a Find a linear equation relating cost to number of books. b If it turns out that the school needs 89 books, what is the cost? A builder can complete a building in das without an assistants, but if he hires four assistants he can build it in das. How man assistants would be required to complete the building in das? A printer quotes a price of $ to print 7 brochures and a price of $ to print brochures. a Find a linear equation relating cost to number of brochures. b Find the cost of printing brochures. A chef can cater for people if she is given hour in preparation time, while if she caters for people she requires hour and minutes. a Find a linear equation relating preparation time to the number of people. b Predict the preparation time required to cater for people. The selling price of a mathematics tetbook is related to the number of pages in the tet. A -page book sells for $ and each additional pages increases the price b $. a Find a linear equation relating the selling price to number of pages. b Predict the cost of a 6-page book. 6 Angela Shster, a barrister, charges $6 per hour for her legal services. a b Find the linear equation relating charge to time. Predict the cost for a -hour legal case.

19 Chapter Linear graphs 9 7 A computer program can be written in das if there are two programmers working on the project, while if there are five programmers it takes onl das. Predict the time it would take si programmers to complete the project. 8 Nile.com, an internet seller of compact disks quotes a shipping cost of $ for CDs and a cost of $ for CDs. a Find a linear equation relating shipping cost to number of CDs. b What would be the shipping cost for CDs? 9 Five students can clean the cafeteria in minutes, while three students would take 7 minutes. How long should it take one student to complete the task? A tai charges a flagfall of $. plus a distance rate of $.8 per kilometre. a Find a linear equation relating cost to distance travelled. b What would be the cost of a. kilometre journe? c How far could ou go for $7.? WorkSHEET. MATHS MATHS QUEST C H A L LL E N G G E E There are eight volumes of a reference series on a bookshelf as shown. The books need to be rearranged so that the are numbered from to 8 going from left to right. If taking a book from the shelf, sliding some books to one side and placing the book back counts as one move, what is the smallest number of moves needed to arrange the series correctl? How man pairs of positive integers for a and b will make the equation a + b = true? What pattern can be observed with the pairs of numbers found in question?

20 9 Maths Quest for Victoria Old geser Old Faithful is a geser in Yellowstone National Park in Woming, USA. For more than ears, Old Faithful has erupted ever da at intervals of less than hours. It is thought that the time between eruptions ma be related to the length of the previous eruption. The following data have been collected. Length of eruption (min) Time until net eruption (min) Your task is to find a linear equation that models the data about Old Faithful. Plot the points on a Cartesian plane with the length of the eruption along the horizontal ais. Rule a straight line which is as close as possible to all of the points. This is called a line of best fit. Write an equation for our line of best fit b choosing two points on the line. Suppose an eruption of Old Faithful lasts. min. Use the equation to predict the time until the net eruption. Data for a geser in New Zealand are also recorded. Length of eruption (min) Time until net eruption (min) Plot the data to find an equation that models this geser. 6 If an eruption from this geser lasts. min, predict the time until the net eruption.

21 Chapter Linear graphs 9 Find the linear equation of a line with a gradient of and a -intercept of. Determine the equation of a linear graph passing through (, ) with a gradient of. Determine the equation of the linear graph passing through (, ) with a gradient of. Find the equation of the straight line shown above right. Find the equation of the straight line shown below right. 6 A straight line passes through the points (, ) and (, ). Find the gradient of this line. 7 Find the equation of the line in question 6. 8 Find the equation of a straight line through the origin with a gradient of. 9 Blue cabs charge $ flagfall plus $. per kilometre. State a linear equation to represent this situation. How far could ou travel in a Blue cab for $? Sketching linear inequations As ou saw in chapter, inequations involve the inequalit signs > (greater than), (greater than or equal to), < (less than) and (less than or equal to). The graph of a linear inequation is actuall a half plane. It is related to the graph of the corresponding linear equation in that the line forms the boundar of the half plane. The convention is to shade the unwanted part of the coordinate plane or region. This can be seen b comparing the graph of the straight line = + with the related graph of the half plane given b < +. Once we have drawn the boundar (linear function) we need to decide which side of the line satisfies the inequalit. We determine this b substituting an point that does not lie on the line into the inequation. If this generates a TRUE statement, it is the half plane that contains (, ) that is required and so the opposite half plane is shaded. If substituting (, ) generates a FALSE statement the half plane containing (, ) is not required and so this side is shaded. If we test (, ) for the inequation < +, we have the statement < + which is true. The half plane containing (, ) is the region required and so the opposite half plane is shaded. A broken line is alwas used for < and > signs as the boundar is not included in the half plane, while a solid line is used for and signs as the boundar is included. = + < + Region required

22 9 Maths Quest for Victoria WORKED Eample Sketch the half plane given b each of the following inequations. a > + 8 b THINK WRITE a Write the inequation. a > + 8 Write the corresponding equation Boundar is = + 8. which forms the boundar. Find the -intercept. -intercept: when =, = + 8 = 8 = -intercept (, ) State the -intercept from the equation -intercept (, 8) as c = 8. 6 For the inequalit >, sketch a broken line passing through both intercepts. Test (, ) and check if the statement Test point: (, ) generated is TRUE or FALSE. > + 8 FALSE 7 The statement is FALSE so the opposite Region required half plane to (, ) is required. Shade > + 8 the half plane containing (, ) and 8 label the other side as the Region required. b Write the inequation. b Write the corresponding equation Boundar is =. which forms the boundar. Find the -intercept. -intercept: when =, = = -intercept (, ) Find the -intercept. -intercept: when =, = = -intercept (, ) For the inequalit, sketch a solid line 6 7 passing through both intercepts. Test (, ) and check whether a TRUE or FALSE statement is generated. The statement is FALSE so the opposite half plane to (, ) is required. Shade the half plane containing (, ) and label the other side as the Region required. Test point: (, ) + FALSE Region required

23 Chapter Linear graphs 9 Graphics Calculator tip! Graphing a half plane You can use our graphics calculator to produce a shaded half plane. Suppose we want to show the half plane shown in worked eample (b).. We need to have the equation in the form = m + c, that is, --.. Press Y= and enter -- for Y.. Before Y ou should see a diagonal line \. This is the line stle to be shown. Press until this is flashing.. We want to shade above the line so keep pressing ENTER (it should be two presses) until an icon for shading above a line can be seen.. Press ZOOM and select 6:ZStandard to show the required half plane. remember remember. Inequations involve the inequalit signs > (greater than), (greater than or equal to), < (less than) and (less than or equal to).. The graph of a linear inequation is a half plane. The corresponding linear equation gives the boundar of the half plane.. The convention is to shade the unwanted part of the coordinate plane or region. Use a test point such as (, ) to check which side of the line to shade.. A broken line is alwas used for < and > signs, while a solid line is used for and signs.. Before sketching the graph of a linear inequation, it ma be necessar to rearrange it to make the subject. 6. When multipling or dividing both sides of an inequation b a negative number, we must reverse the inequalit sign.

24 96 Maths Quest for Victoria E Sketching linear inequations SkillSHEET.6 WORKED Eample a Sketch the half plane given b each of the following inequations. a b < c > d < 6 e > f < g h 8 i j < + k < 7 l WORKED Eample b Sketch the half plane given b each of the following inequations. a + b + < c < -- d e f > 6 g < + h i 6 + > 9 j k + < l Verif our solutions to questions and using technolog. multiple choice a The region satisfing the inequation > is: A B C Region required D E b The region satisfing the inequation + is: A B C Region required D E

25 Chapter Linear graphs 97 c The region satisfing the inequation < is: A B C Region required D E a Sketch the half plane represented b the region: i + ii. b Show the region where both the inequalities + and hold true. 6 Show the region where the inequalities + < and > simultaneousl hold true. Maria s kitchen At the beginning of the chapter we met Maria who was making some high energ biscuits using peanuts and chocolate chips. Maria wanted to make a maimum of g of biscuits but wanted the biscuits to contain at least 8 g of carbohdrates. If we let the mass of peanuts be p and the mass of chocolate chips be c write an inequation to represent the fact that the total mass must be less than g. On a number plane sketch the region defined b the inequation obtained in part. You need to consider onl the positive aes as the values of both p and c must be positive. The peanuts provide % of their mass in carbohdrates and the chocolate chips provide 6% of their mass in carbohdrates. Write an inequation that represents the fact that the mass of carbohdrates must be greater than 8 g. On a number plane sketch the region defined b the inequation obtained in part. On a number plane show the region where the inequalities sketched in parts and both hold true. 6 The region obtained in part shows all possible masses of peanuts and chocolate chips that meet Maria s requirements. List five sets of possible masses of peanuts and chocolate chips that would meet her requirements. WorkSHEET.

26 98 Maths Quest for Victoria summar Cop the sentences below. Fill in the gaps b choosing the correct word or epression from the word list that follows. The is a grid, consisting of two aes ( and ), meeting at the origin (, ). A location (point) is specified b its and. A linear graph consists of an set of points that can be joined to form a straight line. To plot a linear graph, the coordinates of onl two are required although several points are usuall plotted as a check. A linear equation can be epressed in the form, where m represents the gradient (slope) of the straight line and c represents the -intercept. A straight line with a gradient slopes upward as we move in the positive direction while a straight line with a gradient slopes downward as we move in the positive direction. 6 The method allows us to sketch the graph of an linear equation b finding two particular points, the -intercept, and -intercept. 7 Graphs of equations in the form = c have a gradient of zero and are to the -ais. 8 Graphs of equations in the form = a (where a is a constant) have an gradient and are parallel to the -ais. 9 The equation of a straight line can be found b the values of the gradient (m), where m = rise , and -intercept (c) into = m + c. run Linear uses a linear equation to predict a -value from a given -value, or an -value from a given -value. Inequations involve the signs > (greater than), (greater than or equal to), < (less than) and (less than or equal to). The graph of a linear inequation is a. The boundar of the half plane is defined b the corresponding linear equation. The convention is to shade the part of the coordinate plane or region. A line is alwas used for < and > signs while a solid line is used for and signs. When multipling or dividing both sides of an inequation b a negative number, we must the inequalit sign. WORD LIST negative infinite positive unwanted undefined cartesian plane intercept modelling substituting points parallel = m + c coordinates reverse broken half plane inequalit

27 Chapter Linear graphs 99 CHAPTER review Produce a table of values, and sketch the graph of the equation = + for values of between and +. Find the gradient and -intercept of the following straight lines. a = b = -- c = d = Sketch graphs of the following linear equations. a = 6 b + = c + = d + + = Sketch the graph of each of the following. a ( ) = 6( + ) b = c = d = 7 Find the linear equation given the information in each case below. a gradient =, -intercept = b gradient =, -intercept = c gradient = --, -intercept = d gradient =, -intercept = 6 6 For each of the following, find the equation of the straight line with the given gradient and passing through the given point. a gradient = 7, point (, ) b gradient =, point (, ) c gradient = --, point (, ) d gradient = --, point (, ) 7 Find the equations of the straight lines having the following graphs. a b 7 A B B B C C C c d (, 8) 6 e f

28 Maths Quest for Victoria D 8 The cost of hiring a boat is $6 per da plus $. per hour. E a b c Sketch a graph showing the total cost for between and hours. State the equation relating cost to time rented. Predict the cost of hiring a boat for hours and minutes. 9 Sketch the half plane given b each of the following inequations. a + b + c > d < e 7 f -- + g + 9 h 8 i > test ourself CHAPTER