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1 TOPIC Linear relationships. Overview Through understanding linear relationships we can begin to construct and analse linear models. These models can be used to help us see and understand patterns that occur in the world. Linear models can be used to help us solve problems such as the distance covered b an airplane travelling at constant speed, or how much fuel can be purchased with a given amount of mone. DISCUSSION Can ou think of some more real-life situations which can be modelled b linear relationships? Are there an limitations that must be put on these models in order for them to work? LEARNING SEQUENCE. Overview. Direct variation. The gradient and intercepts. Sketching linear graphs. Linear models. Review CURRICULUM CONTENT Students: model, analse and solve problems involving linear relationships, including constructing a straight-line graph and interpreting features of a straight-line graph, including the gradient and intercepts (ACMMM, ACMMM) AAM construct and analse a linear model, graphicall or algebraicall, to solve practical direct variation problems, including but not limited to the cost of filling a car with fuel or a currenc conversion graph (ACMGM) AAM TOPIC Linear relationships

2 . Direct variation.. Defining direct variation Two quantities var directl, or are said to be directl proportional, when one quantit is a constant multiple of the other quantit. An eample of direct variation can be seen in the relationship between the sides of a square and its perimeter. If the side of a square is cm, then the perimeter is cm. Or generall, if the side of the square is cm, then the perimeter, P cm, is times the side. This can be written simpl as P =. As increases, P also increases and is alwas four times the length of the side. This relationship can also be illustrated in a table of values or graphicall. P Since the sides of squares do not have to be integers, the dots can be joined to produce a straight-line graph. A direct variation relationship produces a straight-line graph which passes through the origin. It is a linear function in the form = m, where m is the constant multiple. WORKED EXAMPLE P A special drink is being sold for $ per can. Anita decides to purchase up to cans. Is there a direct variation relationship between the total cost of the cans and the number of cans purchased? If there is a direct variation, determine the relationship. THINK Identif the two quantities and choose a pronumeral to represent them. Use the information to complete a table of values. Remember the cans cost $ each. WRITE = number of cans purchased = total cost of cans can () costs $ (). cans () costs $ (). and so on 7 9 Are both quantities increasing at a constant rate? The number of cans is increasing b. The cost of the cans is increasing b. Answer the question. There is a direct variation between the number of cans and the total cost of the cans. It can be written as =. Jacaranda Maths Quest Mathematics Standard E

3 Note: In Worked eample the number of cans has to be a whole number, so if we drew a graph to represent this situation we could not join the plotted points with a straight line. WORKED EXAMPLE Consider the linear function =. a Complete the table of values for =. b State wh this is a direct variation relationship. THINK WRITE a Substitute the required values of either or When =, = into the linear function =. = When =, = = When =, = = When =, = = When =, = = = When =, = = =. Complete the table.. b For a direct variation relationship, it must be a straight-line graph and pass through the origin (, ). Both values must increase b a constant. WORKED EXAMPLE A straight-line graph, starting at the origin, is shown. a From the graph, generate a table of values to illustrate this information. b Do these points represent a direct variation relationship? c State the relationship between and. d Complete the following table. 9 For =, when = : = = So it passes through point (, ). The -value is alwas times the -value, so as increases, increases. Therefore, = is a direct variation relationship. TOPIC Linear relationships

4 THINK a Enter the given values of and from the graph. Remember the -ais is the horizontal ais and the -ais is the vertical ais... Straight-line graphs passing through the origin If the line formed is a straight line passing through the origin, there is a direct variation relationship between the -coordinates and the -coordinates of the points on the line. Likewise, if a straight-line graph passes through the origin, then it is representing a direct variation relationship. The equation that describes the relationship between the - and -coordinates of the points can be determined b listing the coordinates in a table of values. The table of values for the points on the number plane is as shown. WRITE.. b As increases, does increase? The -values increase b. The -values increase b.. Does the graph pass through the origin? The graph passes through the origin. Answer the question. These points represent a direct variation relationship. c Use the table of values or the graph to find the. =. relationship between and.. =. =.. = The pattern is. = or =.. d Enter the required values of either or into the linear function =.. For = : =. = 9 For = : =.. = = Complete the table The table shows that the -coordinate is alwas twice the value of the -coordinate, so this can be described b the linear function =. All direct variation relationships can be represented b a linear function in the form = m... Constructing straight-line graphs in the form = m Straight-line graphs can be plotted if ou are given: a table of values an two points on the straight line the linear function = m. For = : =. =. For = 9: 9 =. 9. = = Jacaranda Maths Quest Mathematics Standard E

5 Hand hints when constructing straight-line graphs: The horizontal ais is the -ais and the vertical ais is the -ais. A table of values can be ver useful when the linear function is given. Each ais needs to be marked with a suitable scale. Remember, the aes can have different scales, depending on the values to be plotted. Plot each point in the table and join with a straight line. Straight-line graphs can be etended past the given points and will include other points that fit the given equation. There are a number of graphing packages available online or through app stores, often free. Usuall the ask for the relationship and then graph it on a prepared number plane. Following are two eamples of using technolog to construct straight-line graphs. WORKED EXAMPLE With the aid of technolog, graph the straight line = using: a the free online Desmos graphing calculator b Microsoft Ecel. THINK a Follow these steps. Open the Desmos graphing calculator. Click Start Graphing. Using our kepad, enter the equation in line : =. Use the + or on the right-hand side of the graph to zoom in or out. Click on the line to show the coordinates of the points. b Follow these steps. Open an Ecel workbook page. In column A, enter in the first cell. Enter the values of from to. In column B, enter = in the first cell. In cell B enter the formula = *A. Press enter, then drag down the bottom right-hand corner of the cell to complete the column. Highlight values in columns A and B. Insert: Charts: Scatter with Straight Lines and Markers. WRITE 7 RESOURCES Interactivit: Plotting linear graphs (int-) Weblink: Desmos graphical calculator = TOPIC Linear relationships

6 Eercise. Direct variation Understanding, fluenc and communicating. WE A shop has a special offer on DVDs at $ per DVD. Benji decides to purchase up to DVDs and needs to know how man he can afford to purchase. a. Is there a direct variation relationship between the total cost of the DVDs and the number of DVDs purchased? b. If there is a direct variation, determine the relationship.. Biscuits are advertised at $. per packet. Charlotte decides to purchase a number of packets. Determine the relationship between the number of packets purchased, n, and the total cost, $c.. WE Consider the linear function =. a. State wh this is a direct variation relationship. b. Complete the table of values for =, giving values correct to decimal places where necessar.. Consider the linear function =. a. State wh this is a direct variation relationship. b. Complete the table of values for =.. WE A straight-line graph, starting at the origin, is as shown. a. From the graph, generate a table of values to illustrate this information. b. Do these points represent a direct variation relationship? c. State the relationship between and. d. Complete the following table. 7. A straight-line graph is as shown. a. From the graph, generate a table of values to illustrate this information. b. Do these points represent a direct variation relationship? c. State the relationship between and. d. Complete the following table Jacaranda Maths Quest Mathematics Standard E

7 7. a. Given the following tables of values, plot their straight-line graphs on separate number planes. 9 b. Determine the direct variation relationship for these straight-line graphs. Write our answers in the form = m.. For each of the following relationships, complete a table of values from = to =. Represent each of the graphs on a number plane. a. = 9 b. = 7 c. =. 9. WE Using technolog of our own choice, plot the straight-line graphs found in question.. Use technolog of our choice to construct the straight-line graphs of: a. = 7 b. =. Problem solving, reasoning and justification. Anita works casuall in a cafe and is paid $. per hour. She is working out how much she will earn. Let n hours represent the number of hours worked and $c the total amount earned. a. Is there a direct variation relationship between n and c? Justif our answer. b. Create a table of values for n and c, where n =,,,,,. c. Draw the graph of the number of hours worked, n, against the total earned, c. d. Determine the relationship between n and c.. The distance, d kilometres, a fast train travels in t hours is given b the formula d = t. a. Is there a direct variation relationship between d and t? Justif our answer. b. Create a table of values for d and t, where t =,,,,,. c. Draw the graph of distance, d, against time, t.. Below are three tables of values relating and. i ii. 9 iii. 9 TOPIC Linear relationships 7

8 a. From these tables of values, plot the points on separate number planes to represent the relationships. b. For each graph, determine if it is a direct variation relationship. Justif our answer.. The circumference of a circle, C, is given b the formula C = πd, where d is the diameter of the circle. a. Find the circumference, correct to decimal places, of a circle with a diameter of cm. b. Find the circumference, correct to decimal places, of a circle with a radius of cm. c. Is there a direct variation relationship between the circumference of a circle and its diameter? Justif our answer. d. Construct a straight-line graph to represent this relationship.. Andrew is driving from Sunn Beach to Jetville, a distance of 7 km. He drives at a constant speed of 9 km/h. Let n hours represent the number of hours driving and k kilometres the total distance travelled. a. Is there a direct variation relationship between distance travelled and time? Justif our answer. b. Create a table of values for n and k, where n =,,,,,. c. Draw the graph of the number of hours driving, n, against the total distance travelled, k. d. Determine the relationship between n and k. e. How far does Andrew travel in. hours? f. How long does it take Andrew to drive from Sunn Beach to Jetville?. The gradient and intercepts.. Linear relationships A pattern that changes consistentl b the same amount is called a linear pattern. A simple pattern such as,,,, is a linear pattern. A linear relationship means that as one variable changes uniforml, so does the other. Linear relationships can be identified b using a table of values, plotting points or looking at an equation Each variable has a constant difference. = + Both variables are raised to the power of. = and =. = + (, ) (, 9) (, ) (, ) (, ) Points form a straight line. Jacaranda Maths Quest Mathematics Standard E

9 WORKED EXAMPLE For the following tables of values, determine whether a linear relationship eists. Eplain our reasoning. a b c THINK 9 a Look for constant differences in the -values and in the -values. The -values have a constant difference of +, as shown in blue. The -values have a constant difference of, as shown in pink. WRITE Answer the question. As the -values increase b, the -values decrease b. Since is changing b a constant amount and is changing b a constant amount, the table of values represents a linear relationship. b Look for constant differences in the -values and in the -values. The -values have a constant difference of, as shown in blue. The -values do not change b a constant amount Answer the question. As the -values increase b, the -values increase b different amounts each time;, then, then and then. Since is not changing b a constant amount, the relationship is not linear. c Look for constant differences in the -values and in the -values. The -values do not have a constant difference Answer the question. Since the -values increase b different amounts each time, this method cannot be used to determine whether the table represents a linear relationship. TOPIC Linear relationships 9

10 .. The -intercept All linear relationships can be represented b a linear function in the form = m + c, where and are variables and m and c are constants. The graph shows four parallel straight lines. Each line has the same slope but passes through the -ais at a different point. The -intercept is the point at which a line crosses the -ais. As ou can see, the -intercept and the constant in the equation (represented b the letter c) are related. In fact, regardless of the value of m in the equation = m + c, the value of c will alwas be the -value of the -intercept... The -intercept From the previous table, ou can see that ever -intercept has an -value of. The -intercept is where the straight line crosses the -ais, and occurs when =. The -intercept is where the straight line crosses the -ais, and occurs when =. WORKED EXAMPLE State the coordinates of the - and -intercepts in the following graph. -intercept -intercept THINK The -intercept is the point at which the line intersects the -ais. At the -intercept, the -coordinate is equal to. The -intercept is the point at which the line intersects the -ais. At the -intercept, the -coordinate is equal to. = + = + = = Line colour Equation Cuts -ais at: -intercept Blue = + (, ) Pink = + (, ) Green = (, ) Orange = (, ) WRITE -intercept = The -intercept is at (, ). -intercept = The -intercept is at (, ). Jacaranda Maths Quest Mathematics Standard E

11 WORKED EXAMPLE 7 Calculate the - and -intercepts for the graph with the equation = +. THINK To find the -intercept, remember that the -coordinate of the intercept is. Substitute = into the equation, as shown in black, and calculate the value of. To find the -intercept, remember that the -coordinate of the intercept is. Substitute = into the equation, as shown in pink. Calculate the value of b using inverse operations... The gradient = Some lines ma appear as if the are = Line colour Equation Slope going uphill, while others will head = Orange = downhill. The graph at right shows a Pink = number of straight lines with the same -intercept but different slopes. Green = The coefficient of in the equation and Blue = the slope of the line are related. The slope or gradient of a line can be determined b measuring the = change in the -value for each increase of unit in the -value. The change in the -value is referred to as the rise. A negative value for the rise indicates that the -value is decreasing. The increase in the -value is referred to as the run. Gradient = rise run If a right-angled triangle is formed (using the line itself as the hpotenuse), the rise (vertical distance) and run (horizontal distance) can be found. The gradient (m) of the line can be calculated b dividing the rise b the run. m = rise run In the diagram shown at right, the line has a rise of units and a run of unit. Therefore, the slope = rise run = =. is m. When c = the equation. WRITE = + = + = The -intercept is the point (, ). = + = + = = The -intercept is the point (, ). = + units unit The gradient of a straight line with the equation = m + c is the value of the coefficient of, which, this is also known as the constant of variation, as the value of m is a constant in TOPIC Linear relationships

12 A positive gradient has a positive slope. WORKED EXAMPLE Determine the gradient of the line for each of the graphs shown. a b THINK A negative gradient has a negative slope. A horizontal line has no slope and a gradient of zero. A negative slope follows the same slope as the diagonal line in a capital letter N. a Gradient = rise run. Look for two points where the - and -coordinates can be easil read from the grid, as shown in pink. Form a right-angled triangle and use this to find rise and run. The -value increases from to, a rise of. The -value increases from to, a run of. WRITE A vertical line has an infinite slope and an infinite or undefined gradient. egative The greater a slope s magnitude (positive or negative value), the steeper the line formed. m m = rise m = Vertical Horizontal run distance distance rise run = Rise of Run of m = rise run = m = Rise of Run of rise run = m = Fall of Run of rise run = Fall of Run of Jacaranda Maths Quest Mathematics Standard E

13 b Write the gradient formula and calculate the gradient. Gradient = rise run = = Look for two points where the - and -coordinates can be easil read from the grid. Form a right-angled triangle and use this to find rise and run. The -value decreases from to, a rise of. The -value increases from to, a run of. WORKED EXAMPLE 9 Write the gradient formula and calculate the gradient. Gradient = rise run State the gradient and the - and -intercepts of the following linear functions. a = b = c = + d + = e + = THINK WRITE a Write the equation in the form = m + c. = Identif the coefficient of as the gradient. m = Identif the value of c, the -intercept. c = For the -intercept, substitute =. = = Answer the question. For =, the gradient is, the -intercept is (, ) and the -intercept is (, ). b Write the equation in the form = m + c. = Identif the coefficient of as the gradient. m = Identif the value of c, the -intercept. c = For the -intercept, substitute =. = = = Answer the question. For =, the gradient is, the -intercept is (, ) and the -intercept is (, ). = TOPIC Linear relationships

14 c Write the equation in the form = m + c. = + Identif the coefficient of as the gradient. m = Identif the value of c, the -intercept. c = For the -intercept, substitute =. = + = = Answer the question. For = +, the gradient is, the -intercept is (, ) and the -intercept is (, ). d Write the equation in the form = m + c. + = Identif the coefficient of as the gradient. m = Identif the value of c, the -intercept. c = = + = + = + For the -intercept, substitute =. + () = = = Answer the question. For + =, the gradient is the -intercept is (, ) and the -intercept is (, ). e Write the equation in the form = m + c. + = Identif the coefficient of as the gradient. m = Identif the value of c, the -intercept. c = + = = + = + = + For the -intercept, substitute =. () + = + = = = Answer the question. For + =, the gradient is, the -intercept is (, ) and the -intercept is (, )., Jacaranda Maths Quest Mathematics Standard E

15 RESOURCES elesson: The -intercept (eles-7) elesson: The gradient (eles-9) Interactivit: The gradient (int-) Eercise. The gradient and intercepts Understanding, fluenc and communicating. WE For the following tables of values, determine whether a linear relationship eists. Eplain our reasoning. a. b.. For the following tables of values, determine whether a linear relationship eists. Eplain our reasoning. a. b WE State the - and -intercepts for each of the following straight lines. a. b. c.. State the coordinates of the - and -intercepts in the following graphs. a. b. 7. WE7 Calculate the - and -intercepts for the following straight-line graphs. a. = + b. =. Calculate the - and -intercepts for the following straight-line graphs. a. = + b. = TOPIC Linear relationships

16 7. For each of the linear graphs below, state whether the gradient is positive, negative or zero. a. b. c. d.. Use the triangles provided for each straight line graph below to determine the gradients of the lines. Hint: Check the scale on the aes before simpl counting squares. a. b. 9. WE Calculate the gradient for each of the following straight line graphs. a. b. 7. WE9 State the gradient and the - and -intercepts of the following linear functions. a. = b. = + c. = d. + = e. 7 = f. 7 + = Problem solving, reasoning and justification. Cop and complete the following table. Rise Run Gradient a metres metres b metres c metres. a. Match the descriptions given below with their corresponding lines. i. Straight line with a -intercept of (, ) and a positive gradient ii. Straight line with a gradient of iii. Straight line with a gradient of b. Write a description for the unmatched graph. A B C D Jacaranda Maths Quest Mathematics Standard E

17 . Three right-angled triangles have been superimposed on the graph shown. a. Use each of these to determine the gradient of the line. b. Does it matter which points are chosen to determine the gradient of a line? Eplain. c. Describe the shape of the graph.. Which of the following graphs represents a linear function of the form = m + c? a. c. b. d.. For each of the straight-line graphs shown below: i. state the -intercept ii. calculate the gradient iii. write a linear equation to describe the graph. iv. Do an of these graphs represent direct variation? Eplain our reasoning. a. b. c. TOPIC Linear relationships 7

18 . Sketching linear graphs.. Sketching linear graphs from intercepts Hand hints when constructing linear graphs using the ais intercepts: Plot the -intercept on the -ais. Find the -intercept b substituting = into the equation and solve for. Plot the -intercept on the -ais. Find the -intercept b substituting = into the equation and solve for. Join the points with a straight line, etended in both directions. If both intercepts are at the origin, (, ), choose an other -value and calculate the corresponding -value. Plot this second point and join to the origin. Draw arrows at both ends of the line to give an indication of other points that fit the equation. Name the graph with its equation. WORKED EXAMPLE B finding the - and -intercepts, sketch the graph of the equation = +. THINK The -intercept is found b substituting = into the equation. The -intercept is found b substituting = into the equation. Plot these two points on a suitabl scaled Cartesian plane. Join the two intercepts with a straight line, etending the line in both directions. Write the equation of the line. WORKED EXAMPLE Sketch the graph given b the equation =. THINK The -intercept is found b substituting = into the equation. WRITE = () + = + = -intercept: (, ) = + = = -intercept: (, ) = + WRITE = = = The -intercept is the point (, ). Jacaranda Maths Quest Mathematics Standard E

19 The -intercept is found b substituting = into the equation. As both intercepts are at the origin, we have onl one coordinate so far. Choose an other -value, sa =, and substitute this into the original equation to find a second point. Plot these two points on a suitabl scaled Cartesian plane. Join the two points with a straight line, etending the line in both directions... Sketching linear graphs using the gradient and -intercept Hand hints when constructing linear graphs using the gradient and -intercept: Mark the -intercept on the -ais. Another point is found b using the gradient. The gradient is rise run. From the -intercept, move horizontall b the run value, then verticall b the rise value. For eample, for a gradient of, the run is and the rise is. If the gradient is, the rise will be down b. Connect the two points with a straight line. Remember to have arrows on the ends of the line and to name the graph. WORKED EXAMPLE For each of the linear equations below: i state the gradient and -intercept ii sketch the graph of the equation. a. = b. = THINK a i Compare the equation given with the general form of a linear equation: = m + c. The coefficient of is m (the gradient), and the constant c is the -intercept. = = = The -intercept is the point (, ). = = = Another point is (, ). = WRITE For = Gradient (m) = and -intercept (c) =. TOPIC Linear relationships 9

20 ii Construct a set of aes and mark the position of the -intercept. The -intercept is, as shown in black. The gradient is, so rise run =. From the -intercept, rise and run, then mark in a second point, as shown in pink. The two points can now be connected with a straight line. Write the equation net to the line. b i Compare the equation given with the general form of a linear equation: = m + c. Identif m as the gradient and c as the -intercept. ii Construct a set of aes. Mark in the position of the -intercept at, as shown in black. The gradient is, so rise run =. From the -intercept, rise and run, then mark in a second point, as shown in pink. The two points can now be connected with a straight line to form the graph. Write the equation net to the line. RESOURCES elesson: Sketching linear graphs (eles-99) Interactivit: The intercept method (int-) Interactivit: The gradient intercept method (int-9) Eercise. Sketching linear graphs Understanding, fluenc and communicating. WE For each of the linear functions below: i. find the - and -intercepts ii. sketch the graph of the function. a. = + b. = 9 c. = + = For = or ( = + ) Gradient (m) = and -intercept (c) =. = Jacaranda Maths Quest Mathematics Standard E

21 . For each of the straight-line graphs below: i. find the - and -intercepts ii. sketch the graph of the function. a. = + b. = c. = +. WE Sketch the graphs given b the following linear equations. a. = b. = c. =. Sketch the graphs given b the following linear equations. a. = b. = c. =. WE For each of the linear functions below: i. state the gradient and the -intercept ii. sketch the graph of the function. a. = + b. = + c. = d. = e. = + f. =. For each of the linear functions below: i. state the gradient and the -intercept ii. sketch the straight-line graph. a. = + b. = c. = + d. = e. = + f. = 7. Using the gradient and -intercept, sketch the following linear graphs on the same set of aes. Gradient -intercept. a. Sketch the graph of the linear function with a gradient of and -intercept of. b. State the equation of this straight-line graph. 9. a. Sketch the graph of the linear function with a gradient of and a -intercept of. b. State the equation of this straight-line graph.. Using technolog of our choice, sketch the following linear equations on the same number plane. a. = + b. = + c. =. Problem solving, reasoning and justification. a. Sketch the linear equation = : 7 i. using the -intercept and the gradient ii. using the - and -intercepts. b. Compare and contrast the methods and generate a list of advantages and disadvantages for each method. Which method do ou think is best? Wh?. a. Sketch each of the following straight lines on the same set of aes. i. = + ii. = + b. Find the -intercepts for these straight lines. c. What shape is formed b the three ais intercepts? d. Find the area enclosed b the ais intercepts.. a. Using an appropriate method, sketch each of the following straight lines on the same set of aes. i. = + ii. = iii. + = iv. + + = b. What shape is formed b these four lines? c. Write down the coordinates of the vertices (corners) of this shape. TOPIC Linear relationships

22 . Refer to the diagram and answer the following questions. a. Find the gradient of the straight-line graph. b. State the equation that represents this graph. c. Is this a graph of a direct variation relationship? Justif our answer. d. Cop and complete the statement: The -value is times the -value, so the constant of variation is. e. Is there a relationship between the constant of variation and the gradient? Justif our answer.. MC Which sketch below represents the equation =? a. b. (, ) d.. Linear models e. c... Constructing a linear model Practical, everda problems ma often be modelled b linear equations or straight-line graphs. Eamples include: the cost of filling a car with petrol the cost of hiring a tradesman currenc conversion water leaking from a tank distance travelled at a constant speed. To construct a linear model: identif the two variables identif the constant change between the variables, represented b the gradient, m identif the initial starting point or value, represented b c, the -intercept when = write the relationship in the form = m + c sketch the graph of the linear function if necessar. An eample of a linear model is the amount of pa a casual emploee receives when emploed at a rate of $ per hour. For ever hour worked, the emploee is paid $. The two variables are the number of hours worked, hours, and the amount paid, $. The amount paid is increasing b a constant $ ever hour, so m =. The starting point is $, so c =. Jacaranda Maths Quest Mathematics Standard E

23 This gives the linear model =, where is the amount paid, in dollars, and is the number of hours worked. Note: In a linear model we can also use pronumerals other than and to represent our variables. For eample, we could have used n to represent the number of hours and p to represent the amount paid, giving us the linear model p = n. Hours worked.. Solving practical problems Once the linear relationship that models the practical problem has been found, questions can be answered b substituting into the linear equation and solving the equation for the unknown value. Alternativel, information ma be read from a graph. WORKED EXAMPLE A currenc converter shows that an Australian dollar bus.. Eddie is planning a holida to London. a Find a relationship between the Australian dollar and the pound sterling. b How much would Eddie have if he changed $ to pounds sterling? c A da tour from London is advertised for. How much would this be in Australian dollars? d Is this an eample of direct variation? Eplain our response. THINK WRITE a Identif the variables. : amount in Australian dollars to be converted, $ : value in pounds sterling, Identif the constant change and the starting point. Write the relationship between the Australian dollar and the pound sterling. Constant change =. (ever $ converts to.) Starting point = =. b For $, substitute =. =.() = Answer the question. $ is. c To change to AUD substitute = and solve for. =. =... = Answer the question. The da trip would cost Eddie the equivalent of $. d Answer the question. Amount paid ($) 7 The equation is =. so it is an eample of direct variation of the form = m. TOPIC Linear relationships

24 WORKED EXAMPLE The local petrol station is selling fuel for $./litre. Tom has run out of petrol and needs to fill his car. a Identif the relationship between the total cost and the number of litres purchased. b If Tom needs litres for work net week, how much will this cost? c Tom checks his wallet to find onl a $ note. How man litres of fuel can be bu? d Is this a direct variation relationship? Eplain our answer. e Sketch the graph of the linear model. THINK WRITE a Identif the variables. n: number of litres of fuel purchased C: cost of fuel in dollars Identif the constant change and the starting Ever etra litre costs $.. point. Constant change =. Starting point = $ Write the relationship as an equation. C =.n b For litres, substitute n =. C =. =.7 Answer the question. litres would cost $.7. c For $, substitute C = and solve. C =.n =.n. =.n. n = Answer the question. $ would purchase litres of fuel. d Answer the question. The equation C =.n is an eample of direct variation since it is of the form = m. e State the gradient and -intercept. Gradient =. -intercept =, so plot another point (, ). Plot the two points and join them with a straight C line, or choose technolog to graph. n Jacaranda Maths Quest Mathematics Standard E

25 .. The limitations of models Straight-line graphs have arrows at each end to indicate that the line continues. In modelling practical problems with straight-line graphs or linear functions, limitations ma need to be placed on the variables. For eample, in Worked eample, since and represent mone both and must be zero or positive. Practical problems involving variables such as time, length and volume, which can onl be zero or positive, will have limitations placed on them for the linear model to be valid. WORKED EXAMPLE A water tank has developed a leak, with water flowing out at the rate of litres per minute. Initiall, the water tank contained litres. a Construct a linear model that represents the amount of water, W litres, in the tank an time t minutes after the water started leaking. b Determine how man litres of water remained in the tank after minutes. c Determine how long it would take for the water in the tank to reduce to litres. d If the water continued flowing out, how long would it take for the water tank to be empt? e What are the limitations required for this linear model to be valid? f Sketch the graph of this linear model. THINK WRITE a Use pronumerals to represent the given Let t = time in minutes from when water starts leaking information. Let W = amount of water, in litres, in the tank at an time Find the constant change and the initial Constant change = (negative since flowing out, information. decreasing) Initiall: litres in the tank Write the linear equation. W = t + b Substitute t = into the linear equation. W = + = + = 7 Answer the question. There are 7 litres in the water tank after minutes. c Substitute W = into equation. = t + Solve for t. = t + = t = t. = t Answer the question. It would take. minutes to reduce to litres of water. d To be empt, W =. = t + Substitute and solve for t. t = t = t = Answer the question. It would take minutes for the water tank to empt. TOPIC Linear relationships

26 e Time and capacit both have to be greater than or equal to zero. The model is valid from to minutes, or t and from (empt) to litres. W f Construct the straight line representing W = t +. This is a linear function of the form = m + c, with gradient = and -intercept =. Label the aes. Eercise. Linear models Understanding, fluenc and communicating. WE Maddie is planning a holida to Europe. A currenc converter shows that one Australian dollar bus.7. a. Find the relationship between Australian dollars and euros. b. How much would Maddie have if she changed $ to euros? c. A da tour from Paris is advertised for. How much would this be in Australian dollars? d. Is this an eample of direct variation? Eplain our response.. A currenc converter is showing one Australian dollar would bu Japanese en. Alan is planning a skiing trip to Japan. a. Find the relationship between the Australian dollar and the Japanese en. b. How much would Alan have if he changed $ to Japanese en? c. A weekl ski pass at the resort costs en. How much, in Australian dollars, would Alan have to budget for this? d. Is this an eample of direct variation? Eplain. e. Sketch the linear model relating Australian dollars and Japanese en.. WE The local petrol station is selling fuel for $./litre. Bett has run out of petrol and needs to fill her car with fuel. a. Identif the relationship between the total cost and the number of litres purchased. b. Bett knows her petrol tank holds litres. What would it cost Bett to fill the tank? c. If Bett onl has $ in her purse, how man litres of fuel can she bu? d. Is this a direct variation relationship? Eplain our answer. e. Sketch the graph of the linear model. W t Time (minutes). Diesel fuel is being advertised for. cents/litre. Terr needs to refuel his van. a. Identif the relationship between the total cost and the number of litres purchased. b. If Terr needs litres for the weekend, how much will this cost? c. Terr checks his wallet and finds onl a $ note. How man litres of diesel can be bu? d. Is this a direct variation relationship? Eplain our answer. e. Sketch the graph of the linear model. Amount of water (litres) Jacaranda Maths Quest Mathematics Standard E

27 . WE A water tank has developed a leak, with water flowing out at the rate of litres per minute. Initiall, the water tank contained litres. a. Construct a linear model that represents the amount of water, W litres, in the tank an time t minutes after the water started leaking. b. Determine how man litres of water remained in the tank after minutes. c. Determine how long it would take for the water in the tank to reduce to litres. d. If the water continued flowing out, how long would it take for the water tank to be empt? e. What are the limitations required for this linear model to be valid? f. Sketch the graph of this linear model.. Water is being pumped into a holding tank at a rate of litres/minute. Initiall the holding tank was empt. a. Construct a linear model that represents the amount of water, W litres, in the tank at an time t minutes after starting the pump. b. Determine how man litres of water are in the holding tank after minutes. c. The holding tank has a capacit of litres. Determine the time taken to fill the tank. d. Is this a direct variation relationship? Eplain our answer. e. What are the limitations required for this linear model to be valid? f. Sketch the graph of this linear model. 7. Australia measures speed in kilometres per hour and the United Kingdom in miles per hour. km/h is equivalent to. mph. Ton has been asked to construct a linear model so he and his friends can check their speeds. Ton decides to let represent km/h and mph. a. Construct the linear equation that connects km/h and mph. b. Ton knows the speed limit on the open road in Australia is km/h. What would this be in the UK? c. Ton noticed a road sign in the UK that showed. What would this speed be in Australia? d. Is the situation an eample of direct variation? Eplain.. A jogger runs for. hours. The graph shows the distance, d km, the jogger travels in this time. Use the graph to answer the following questions. a. How far did the jogger run in the first hour? b. Calculate the gradient of the straight-line graph. c. Write an equation to represent the relationship between distance jogged, d km, and time, t hours. d. How man minutes would it take for the jogger to pass the.. -kilometre mark? Time (hours) e. Does this represent a direct variation relationship? Eplain our answer. Distance jogged (km) TOPIC Linear relationships 7

28 9. The graph shows how the cost of a school formal varies with the number of students attending. a. Is there a direct variation relationship between the cost of the formal and the number of students attending? b. Calculate the gradient of the straight-line graph. c. Write the equation that connects the cost, $C, and the number of students, n, attending. d. State the constant of variation. e. Find the total cost if students attended to school formal.. The graph shows the speed of travel, s km/h, versus the tachometer reading, r revolutions/minute, in top gear. a. Is there a direct variation relationship between the speed of travel and the tachometer reading? b. Calculate the gradient of the straight-line graph. c. Write the equation that connects the speed, s km/h, and the tachometer reading, r revolutions/minute. d. State the constant of variation. e. Find the speed of travel for a tachometer reading of revolutions/minute. f. Find the tachometer reading if the speed of travel was km/h. 9 7 Number of people Problem solving, reasoning and justification Tachometer reading (revolutions/minute). The manager of a manufacturing compan purchased a new machine valued at $. The value of the machine depreciates (decreases) constantl over a -ear period. The value of the machine, $v, at an time, t months, is given b the equation v = t. a. Sketch the graph of this relationship. b. What is the value of the machine after ears? c. How long would it take for the machine to halve in value? d. Are there an limitations on the straight-line graph in this contet? Justif our answer. e. Does this situation represent a direct variation relationship? Justif our answer.. Nick is saving part of his pocket mone to bu a new TV. The graph shows his total savings, $, for weeks. a. Construct an equation of this graph, connecting $ and weeks. b. How much would Nick save in a ear? c. The TV is advertised for $. How long will it take Nick to save to bu this TV? Weeks d. Since the graph is increasing b a constant value each week, it represents a direct variation relationship. True or false? Justif our answer.. The fuel-consumption rate of a particular car is given as. km/l. a. Write an equation connecting the distance travelled, D km, with the amount of fuel consumed, n L. b. Calculate the distance travelled on litres of fuel. c. Calculate the amount of fuel required for a trip of km. d. Sketch a graph of the relationship between distance and the fuel consumed. e. Does this represent a direct variation relationship? Justif our answer. f. Complete the statement: Since the gradient of the straight-line model is, then the of variation is. Cost ($) Speed of travel (km/h) 7 Speed of travel versus tachometer reading in top gear Savings ($) Jacaranda Maths Quest Mathematics Standard E

29 . Abigail, Bruce and Charlotte are friends. The are discussing their A wages for working up to hours per da. The graph shows how B their wages compare. C a. Whose wage represents a direct variation relationship? Justif our answer. b. What is Abigail s wage if she onl works hours? c. Construct an equation that connects Abigail s wage, $, with the 7 number of hours worked, hours. Time worked per da (h) d. How man hours would Bruce and Charlotte have to work in a da to receive the same wage? e. Construct an equation that connects Charlotte s wage, $, with the number of hours she works, hours. f. How much more than Charlotte would Abigail make if F the both work for 7. hours?. a. The graph shown is used to convert temperature from degrees Celcius to degress Fahrenheit. Use the graph to convert i. C to degrees Fahrenheit ii. F to degrees Celcius. b. The linear model that converts degrees Celcius to degrees Fahrenheit is given b F = 9 C +. i. Use the linear model to convert: I. C to degrees Fahrenheit II. F to degrees Celcius. ii. Compare the answers from b with those in a. Comment on the results. iii. State the gradient of the straight line shown in part a. iv. Is this an eample of direct variation? Justif our answer.. Review Fahrenheit ( F) Total wage ($) C Celcius ( C).. Summar In this topic ou have learnt: when direct variation occurs about the graphs of direct variation relationships about the gradient of a straight-line graph how to plot points from a table of values how to draw a straight-line graph using two points how to draw a straight-line graph using the gradient and -intercept how to draw a straight-line graph using the - and -intercepts how to draw a straight-line graph with the aid of technolog the general equation for straight-line graphs how to calculate the gradient how to determine a second point using the gradient how to calculate the -intercept how to calculate the -intercept how to create a linear model the limitations that ma need to be placed on variable for the model to be valid. TOPIC Linear relationships 9

30 Eercise. Review Understanding, fluenc and communicating. Given the following tables of values, plot the straight-line graphs on separate number planes. a. c. 9 b. d. 7. For each of the following linear relationships, complete a table of values for =,,,,. Sketch the graphs on separate number planes. a. = b. = + c. = + d. =. + e. =. +. f. =. For each of the straight-line graphs shown below: i. state the -intercept ii. calculate the gradient iii. write a linear equation to describe the graph. a. b.. Three straight-line graphs are shown. For each graph, find: i. the -intercept ii. the gradient iii. the equation of the straight-line graph. a. b. c. c.. State the value of m and c in each of the following linear equations. a. = + b. = + c. = d. = + e. = f. = Jacaranda Maths Quest Mathematics Standard E

31 . Write the linear equations for the lines with the following properties: a. gradient = -intercept = d. gradient = -intercept = b. gradient = -intercept = e. gradient = -intercept = 7. Use the -intercept and gradient to sketch the following straight-line graphs. a. = b. =. Calculate the - and -intercepts of the following linear graphs. a. = + b. = + 7 c. = d. = + e. = f. = 9. Calculate the - and -intercepts of the following linear graphs. a. = b. = + c. =. + d. = + e. = f. = +. For each of the linear equations, calculate the - and -intercepts and sketch the graph. a. + = b. + = c. = d. 7 + = e. + = f. 7 =. For each of the following linear equations: i. rearrange in the form = m + c ii. state the gradient iii. state the -intercept. a. = + b. = c. = +. Sketch the straight-line graphs given b the following equations. a. = + b. = c. = +. d. =. e. = +. f. =.. Sketch the straight-line graphs given b the following equations. a. = b. = c. = d. = e. = f. =.. Sketch the straight-line graphs given b the following equations. a. + = b. = c. + = d. 7 = e. + = f. + =. a. Sketch the straight-line graph with the gradient of. and the -intercept of. b. Sketch the straight-line graph with the gradient and -intercept of.. The wages earned b Tiffan for different numbers of hours are shown in the table. Time (hours) Wages ($) 7 a. How much does Tiffan earn per hour? b. Is there a direct variation relationship between Tiffan s wage and the number of hours she works? c. Determine the relationship. c. gradient = -intercept = f. gradient =. -intercept = TOPIC Linear relationships

32 7. The equation of the straight-line graph shown is: a. = b. = c. = + d. = + e. = +. The gradient of the linear function + = is: a. b. c. d. e. 9. The - and -intercepts for the straight-line graph = are respectivel: a. (, ) and (, ) b. (, ) and (, ) c. (, ) and (, ) d. (, ) and (, ) e. (, ) and (, ). For the straight-line graph = +, which statement is not correct? a. The gradient is. b. The -intercept is. c. When =, = d. It represents a direct variation relationship. e. When =, =. Problem solving, reasoning and justification. a. Plot the graph for each of the following equations from the table of values provided. i. = ii. = b. What do ou notice about the graphs in part a? c. State the gradient of the straight-line graph. d. Is there a reason for the shape of these graphs? Justif our answer.. a. Plot the graph for each of the following equations from the table of values provided. i. = ii. = b. What do ou notice about the graphs in part a? c. State the gradient of the straight-line graph. d. Is there a reason for the shape of these graphs? Justif our answer.. The time, T minutes, that a Year student is epected to take to complete a Maths skill sheet consisting of questions is given b the linear model T =.Q +, where Q is the number of questions the student answers. a. How long is it epected to take students to complete all of the questions? b. State an limitations of this linear model. c. Sketch a graph of time taken, T, for the number of questions answered, Q. d. What is a possible eplanation of the number in the given equation. e. Is this a linear model of a direct variation relationship? Give reasons for our answer. Jacaranda Maths Quest Mathematics Standard E

33 . The graph shows the connection between the number of revolutions of a biccle wheel and the distance travelled in metres. a. Does this graph represent a direct variation relationship between the distance travelled and the number of revolutions? Give reasons for our answer. b. Calculate the gradient of the straight line. c. Write an equation in the form = m, where represents the distance travelled and the number of revolutions. d. Calculate the distance travelled, in kilometres, for revolutions of a biccle wheel. e. Calculate the number of revolutions required to travel a distance of. kilometres.. The linear model that converts degrees Celcius to degrees Fahrenheit is given b F = 9 C +. a. The average maimum and minimum temperatures for four cities are given in the following table. Cit Maimum temperature Minimum temperature Sdne, NSW C C London, UK 9 C C New York, USA 7 F F Los Angeles, USA 7 F F i. Which cit has the lowest minimum temperature? Justif our answer. ii. What is the difference between the average maimum temperatures of London and Los Angeles? Give our answer in degrees Fahrenheit. iii. Convert the New York temperatures to degrees Celcius, correct to decimal places. b. An old cake recipe shows an oven setting of F. Your oven is in degrees Celcius. What temperature would ou set our oven to? Give our answer to the nearest degrees.. The local petrol station is showing fuel for 9.9 cents/l. Tom has run out of petrol and needs to fill his car. a. Identif the relationship between the total cost and the number of litres purchased. b. Tom knows his petrol tank holds 7 litres. What would it cost Tom to fill the tank? c. Is this a direct variation relationship? Eplain our answer. d. Sketch the graph of the linear model. e. State an limitations of this linear model. Number of revolutions 7. The fuel-consumption rate of a new car was advertised as. km/l. a. Write an equation connecting the distance travelled, D km, with the amount of fuel consumed, n litres. b. The fuel tank holds L. If the fuel tank is full, how far are ou able to travel? c. A road trip of km is being planned. How man litres of fuel would ou epect to use? d. Sketch a graph of the relationship between distance and the fuel consumed. Distance travelled (m) TOPIC Linear relationships

34 e. Does this represent a direct variation relationship? Justif our answer. f. Complete the statement: Since the gradient of the straight-line model is, then the of variation is.. A tank has developed a leak, with water flowing out at the rate of litres per minute. Initiall, the water tank contained litres. a. Construct a linear model that represents the amount of water, W litres, in the tank an time t minutes after the water started leaking. b. Sketch the graph of this linear model. c. Determine how man litres of water remained in the tank after an hour. d. If the water continued flowing out, how long would it take for the water tank to be empt? Give our answer in hours and minutes. e. What are the limitations required for this linear model to be valid? f. Is this a model of a direct variation relationship? Eplain. 9. Fuel is being pumped into a petrol tank at a rate of L/min. Initiall the holding tank was empt. a. Construct a linear model that represents the amount of fuel, F litres, in the tank at an time t minutes after starting the pump. b. The capacit of the petrol tank is litres. Determine the time taken to fill the tank. Give our answer in hours and minutes to the nearest minute. c. Sketch the relationship of the linear model. d. What are the limitations required for this linear model to be valid? e. Is this a model of a direct variation relationship? Eplain.. A currenc converter is showing one Australian dollar would bu. Indian rupees. Ben is a cricket fan and plans to fl to India for a test match. a. Find a relationship between Australian dollars and Indian rupees. b. How much would Ben have if he changed $ to Indian rupees? c. Tickets for a match range from to rupees. Find this range in Australian dollars. d. Is this an eample of direct variation? Eplain. e. Sketch the linear model relating Australian dollars and Indian rupees.. Charlie has spent the da at the pool. When he was leaving the pool to drive home, he realised there was a linear model that connected his distance from home, d km, with the time he takes to get home, t minutes. The equation was d = t. a. Sketch the graph of this equation. b. How far is the pool from Charlie s home? c. How long does it take for Charlie to drive home? d. What are the limitations on this linear model? e. Does this linear model represent a direct variation relationship? Justif our answer. Jacaranda Maths Quest Mathematics Standard E