SAMPLE. Interpreting linear relationships. Syllabus topic AM2 Interpreting linear relationships. Distance travelled. Time (h)

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1 C H A P T E R 5 Interpreting linear relationships Sllabus topic AM Interpreting linear relationships Graphing linear functions from everda situations Calculating the gradient and vertical intercept Using and interpreting graphs of the form = m + b Solving simultaneous linear equations from a graph Using linear functions to model and interpret practical situations 5. Graphing linear functions A linear function makes a straight line when graphed on a number plane. There are man everda situations that result in a linear function such as the distance travelled as a function of the time (d = 5t). The graph of d = 5t is shown below. 5 5 d Distance travelled Distance (km) Time (h) t 5 Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 5 8/9/ 9: PM

2 6 Preliminar Mathematics General Independent and dependent variables The linear function d = 5t has two variables d (distance) and t (time). Time (t) is the independent variable, as an number can be substituted for this variable such as t =. Conversel, the distance (d) is the dependent variable is it depends on the number substituted for the independent variable. That is, when time is (t = ) then the distance is d = 5 or 5. Graphing a linear function Construct a table of values with the independent variable as the first row and the dependent variable as the second row. Draw a number plane with the independent variable on the horizontal ais and the dependent variable as the vertical ais. Plot the points. Join the points to make a straight line. Eample Graphing a linear function from a table of values The table below shows the cost of postage (c) as a function of the weight of the parcel (w). a b Weight (w) 5 Cost (c) Draw a graph of cost (c), against the weight of the parcel (w). Use the graph to determine the cost of a parcel if the weight is.5 kg. Solution Draw a number plane with the weight of parcel (w) as the horizontal ais and the cost of postage (c) as the vertical ais. Plot the points (,.), (,.), (,.6), (,.8) and (5, 6.). Join the points to make a straight line. Find.5 kg on the horizontal ais and draw a vertical line. Where this line intersects the graph, draw a horizontal line to the vertical ais. 5 Write the answer in words. a Cost of postage in $ c Cost of postage 6 5 w 5 Weight of parcel in kg b.5 kg would cost about $. Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 6 8/9/ 9: PM

3 Chapter 5 Interpreting linear relationships 7 Eample Graphing a linear function Draw the graph of = -. Solution Draw a table of values for and. Let = -, -,, and. Find using the linear function = -. Draw a number plane with as the horizontal ais and as the vertical ais. Plot the points (-, -5), (-, -), (, -), (, ) and (, ). 5 Join the points to make a straight line Eample Graphing a linear function using a graphics calculator Use a graphics calculator to draw the graph of = -. Solution Select the Graph menu. Enter the formula = - b tping X at Y. The graph of Y = X is the same as = -. Edit the aes to an appropriate scale. Select SHIFT F for the V-Window. Enter the Xmin =, Xma =, Ymin = 5, Yma =. 5 Press EXE to eit V-Window. 5 6 Select F6 to draw the graph. Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 7 8/9/ 9: PM

4 8 Preliminar Mathematics General Eercise 5A Chocolates are sold for $ per kg. The table below shows weight against cost. Weight (w) 5 Cost (c) a Which is the dependent variable? b Which is the independent variable? c Draw a graph of weight against cost. d Use the graph to find c if w is.5. Mobile phone call costs are charged at a rate of cents per minute. Time (t) 5 6 Cost (c) a Which is the dependent variable? b Which is the independent variable? c Draw a graph of time against cost. d Use the graph to find t if c is.9. Soraa conducted a science eperiment and presented the results in a table. Mass (m) Time (t) a Draw a graph of mass against time. b Use the graph to find t if m is. Complete the following table of values for each linear function. a = + c b = a + a - - b b = 6 8 d q = -p + p Use the table of values from the above question to graph these linear functions. a = + b = c b = a + d q = -p + q Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8 8/9/ 9: PM

5 Chapter 5 Interpreting linear relationships 9 Development 6 The cost (c) of apples is $.5 per kilogram and is determine b the formula c =.5w where (w) is the weight of the apples. a Construct a table of values for the weight against cost. Use,,, and for w. b Draw the graph of the weight (w) against the cost (c). c How man kilograms of apples can be purchased for $5? 7 The age of a computer (t) in ears to its current value (v) in $ is v = -5t +. a Construct a table of values for the age against current value. (t =,,,, ) b Draw the graph of the age (t) against current value (v). c What is the initial cost of the computer? d What will be the current value of the computer after two ears? e When will the computer be half its initial cost? 8 The cost of hiring a tai is $ flag fall and $ per kilometre travelled. a Construct a table of values using,,, and as values for kilometres travelled (d) and calculating cost of the tai (C). b Draw the graph of the kilometres travelled (d) against cost of tai (C). 9 Emil was born on Jack s tenth birthda. a Construct a table of values using,,, and as values for Emil s age (E) and calculating Jack s age (J). b Draw the graph of the Emil s age (E) against Jack s age (J). One Australian dollar (AUD) was converted for. New Zealand dollars (NZD). a Construct a table of values using,,, and as values for AUD and calculate the NZD using the above conversion. b Draw the graph of the AUD against NZD. Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 9 8/9/ 9: PM

6 Preliminar Mathematics General 5. Gradient and intercept Gradient The gradient of a line is the slope or steepness of the line. It is calculated b dividing the vertical rise b the horizontal run. The larger the gradient, the steeper the slope. The letter m is often used to indicate gradient. Horizontal run Vertical rise Vertical rise Gradient (or m) = Horizontal run Positive gradients are lines that go up to the right or are increasing. Conversel, negative gradients are lines that go down to the right or are decreasing. + Positive gradient Eample Finding the gradient of a line Negative gradient Find the gradient of a line through the points (, ) and (, ). Solution Draw a number plane with as the horizontal ais and as the vertical ais. Plot the points (, ) and (, ). Draw a line between the two points. Construct a right-angled triangle b drawing a vertical and a horizontal line. 5 The line is positive as it slopes towards the right. 6 Determine the vertical rise ( = ). 7 Determine the horizontal run ( = ). 8 Substitute for the vertical rise and for the horizontal run into the formula. (, ) Horizontal run (, ) Vertical rise Vertical rise Gradient or m = Horizontal run = + Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8/9/ 9: PM

7 Chapter 5 Interpreting linear relationships Intercept The intercept of a line is where the line cuts the aes. The intercept on the vertical ais is called the -intercept and is denoted b the letter b. The intercept on the horizontal ais is called the -intercept and is denoted b the letter a. Gradient Gradient of a line is the slope of the line. Vertical rise Gradient (or m) = Horizontal run Eample 5 Finding the gradient and vertical intercept Find the gradient and vertical intercept for the line = - +. Solution Draw a table of values for and. Let =, and. Find using the linear function = - +. Draw a number plane with as the horizontal ais and as the vertical ais. Plot the points (, ), (, ) and (, ). 5 Draw a line between these points. 6 Construct a right-angle triangle b drawing a vertical and a horizontal line. 7 The line is negative as it slopes towards the left. 8 Determine the vertical rise ( = ). 9 Determine the horizontal run ( = ). Substitute for the vertical rise and for the horizontal run into the formula. Evaluate. The line cuts the vertical ais at. Intercept The intercept of a line is where the line cuts the aes. Vertical intercept is often denoted b b. - - Vertical rise Vertical intercept Horizontal run Vertical rise Gradient or m = Horizontal run = = Intercept on the vertical ais is. Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8/9/ 9: PM

8 Preliminar Mathematics General Eercise 5B Find the gradient of the following lines. a c 8 8 b d 6 8 What is the gradient of the line that joins these points? a (, ) and (, 5) b (, ) and (, ) c (, ) and (, ) What is the intercept on the vertical ais for the following lines? a b Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8/9/ 9: PM

9 Chapter 5 Interpreting linear relationships Development Plot the following points on number plane and join them to form a straight line. Determine the gradient and -intercept for each line. a c b d Draw a graph of these linear functions and find the gradient and -intercept. a = + b = - + c = + d = e + = f - = 6 The distance (d) a train travels in kilometres is calculated using the formula d = 5t where (t) is the time taken in hours. a Construct a table of values using,,, and as values for t. Calculate the distance (d). b Draw the graph of the distance (d) against the time (t). c What is the gradient of the graph? d What is the intercept on the vertical ais? 7 Meat is sold for $6 per kilogram. a Construct a table of values using,,, and as values for the number of kilograms (n). Calculate the cost (c) of the meat. b Draw the graph of the cost (c) against the number of kilograms (n). c What is the gradient of the graph? d What is the intercept on the vertical ais? Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8/9/ 9: PM

10 Preliminar Mathematics General 5. Gradient intercept formula When the equation of a straight line is written in the form = m + b it is called the gradient intercept formula. The gradient is m, the coefficient of, and the -intercept is b, the constant term. The independent variable in the formula is and the dependent variable in the formula is. The gradient intercept formula is useful in modelling relationships in man practical situations. However, the variables are often changed to reflect the situation. For eample, the formula c = 5n + has c as the cost of the event ($) and n as the number of guests. These letters are the dependent and independent variables. Gradient intercept formula Linear equation = m + b. m Slope or gradient of the line (vertical rise over the horizontal run). b -intercept. Where the line cuts the -ais or vertical ais. Eample 6 Finding the gradient and -intercept from its equation Write down the gradient and -intercept from each of the following equations. a = b = 8 - c = 6 d - = Solution Write the equation in gradient intercept form. Gradient is the coefficient of. -intercept is the constant term. Write the equation in gradient intercept form. 5 Gradient is the coefficient of. 6 -intercept is the constant term. a = = 5 - Gradient is 5, -intercept is b = 8 - = Gradient is, -intercept is 8 Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8/9/ 9: PM

11 Chapter 5 Interpreting linear relationships 5 7 Write the equation in gradient intercept form. 8 Gradient is the coefficient of. 9 -intercept is the constant term. Write the equation in gradient intercept form. Gradient is the coefficient of. -intercept is the constant term. Sketching graphs of linear functions c = 6 = 6 + Gradient is 6, -intercept is d - = = + Gradient is, -intercept is Sketching a straight-line graph requires at least two points. When an equation is written in gradient intercept form, one point on the graph is immediatel available: the -intercept. A second point can be quickl calculated using the gradient or b substituting a suitable value of into the equation. Eample 7 Sketching a straight-line graph from its equation Draw the graph of = +. Solution Write the equation in gradient intercept form. Gradient is the coefficient of or. -intercept is the constant term or. Mark the -intercept on the -ais at (, ). 5 Gradient of (or ) indicates a vertical rise of and a horizontal run of. 6 Start at the -intercept (, ) and draw a horizontal line, unit in length. Then draw a vertical line, units in length. 7 The resulting point (, ) is a point on the required line. 8 Join the points (, ) and (, ) to make the straight line. = + Gradient is, -intercept is (, ) (, ) Rise = Run = Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 5 8/9/ 9: PM

12 6 Preliminar Mathematics General Eercise 5C Write down the gradient and -intercept from each of the following equations. a = + b = - 7 c = 5 +. d =.5 - e = + f = 5 - g = h = + 5 Write down the equation of a line that has: a gradient = and -intercept = b gradient = and -intercept = c gradient = and -intercept = d gradient =.5 and -intercept = Find the equation of the following line graphs. a c b d 8 A straight line has the equation = +. a What are the gradient and the -intercept? b Sketch the straight line on a number plane using the gradient and -intercept. Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 6 8/9/ 9: PM

13 Chapter 5 Interpreting linear relationships 7 Development 5 It is known that varies directl with. When =, =. a Write a linear equation in the form = m to describe this situation. b Draw the graph of against. 6 Kalina s pa (p) is directl proportional to the number of hours (h) she works. For an 8-hour da she receives $68. a Write a linear equation in the form p = m to describe this situation. b Draw the graph of p against h. 7 A bike is travelling at constant speed. It travels 5 km in 7 hours. a Write a linear equation in the form d = mt to describe this situation. b Draw the graph of d against t. 8 Sketch the graphs of the following equations on the same number plane. a = b = + c = + d = - e = - - f = - - g What do ou notice about these graphs? 9 Sketch the graphs of the following equations on the same number plane. a = + b = + c = + d = - - e = - - f = - - g What do ou notice about these graphs? Sketch the graphs of the following equations using the gradient intercept formula. a = + b =.5 - c = d = e + = 5 f + = 8 g = h + = i - = - Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 7 8/9/ 9: PM

14 8 Preliminar Mathematics General 5. Simultaneous equations Two straight lines will alwas intersect unless the are parallel. The point at which two straight lines intersect can be found b sketching the two graphs on the one set of aes and reading off the coordinates at the point of intersection. When the point of intersection is found it is said to be solving the equations simultaneousl. Solving two linear equations simultaneousl from a graph Draw a number plane. Graph both linear equations on the number plane. Read the point of intersection of the two straight lines. Eample 8 Finding the solution of simultaneous linear equations B drawing their graphs find the simultaneous solution of = + and = -. Solution Use the gradient intercept form to determine the gradient and -intercept for each line. Gradient is the coefficient of. -intercept is the constant term. Draw a number plane. 5 Sketch = + using the -intercept of and gradient of. 6 Sketch = - using the -intercept of and gradient of. 7 Find the point of intersection of the two lines (, ). 8 Simultaneous solution is the point of intersection. = + Gradient is, -intercept is = - Gradient is, -intercept is Simultaneous solution is = - and = = + = Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8 8/9/ 9: PM

15 Chapter 5 Interpreting linear relationships 9 Eercise 5D What is the point of intersection for each of these pairs of straight lines? a c = + = =.5 + = b d 5 = + = + = = + Plot the following points on a number plane and join them to form two straight lines. What is the point of intersection of these straight lines? Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 9 8/9/ 9: PM

16 Preliminar Mathematics General Plot the following points on a number plane and join them to form two straight lines. What is the point of intersection of these straight lines? The graph opposite shows the cost of producing boes of sweets and the income received from their sale. a Use the graph to determine the number of boes which need to be sold to break even. b How much profit or loss is made when boes are sold? c How much profit or loss is made when bo is sold? d What are the initial costs? 5 The graph opposite shows the cost of producing a pack of batteries and the income received from their sale. a Use the graph to determine the number of packs which need to be sold to break even. b How much profit or loss is made when 5 packs are sold? c How much profit or loss is made when packs are sold? d What are the initial costs? Dollars ($) Income Costs 5 Boes Income Dollars ($) Costs 5 5 Packs Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8/9/ 9: PM

17 Chapter 5 Interpreting linear relationships Development 6 Draw the graphs of the following pairs of equations and find their simultaneous solution. a = + and = - b = - and = + c = 5 + and = - 7 d = and = + 7 Zaina bus and sells books. Income received b selling a book is calculated using the formula I = 6n. Costs associated in selling a book are calculated using the formula C = 8n +. a Draw the graph of I = 6n and C = 8n + on same number plane. b What are the initial costs? c Use the graph to determine the number of books needed to be sold to break even. d How much profit or loss is made when 6 books are sold? 8 Am and Nghi work for the same compan and their wages are a and b respectivel. a Am earns $ more than Nghi. Write an equation to describe this information. b The total of Am s and Nghi s wages is $5. Write an equation to describe this information. c Draw a graph of the above two equations on the same number plane. Use a as the horizontal ais and b as the vertical ais. d Use the intersection of the two graphs to find Am s and Nghi s wage. 9 A factor produces items whose costs are $ plus $ for ever item. The factor receives $5 for ever item sold. a Write an equation to describe the relationship between the: i costs (C) and number of items (n) ii income (I) and number of items (n) b Draw a graph to represent the costs and income for producing the item. c How man items need to be sold to break even? Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8/9/ 9: PM

18 Preliminar Mathematics General 5.5 Linear functions as models Linear modelling occurs when a practical situation is described mathematicall using a linear function. For eample, the gradient intercept form of a straight-line graph can sometimes be used to model catering costs. A catering compan charges a base amount of $ plus a rate of $5 per guest. Using this information, we can write down a linear equation to model the cost of the event. Let c be the cost of the event ($) and n be the number of guests, we can write c = 5n +. Note: The number of guests (n) must be greater than zero and a whole number. The graph of this linear model has been drawn c Catering cost opposite. There are two important features of this linear model: 5 Gradient is the rate per guest or $5. 5 The c-intercept is the base amount or $. n Eample 9 Using graphs to make conversions The graph opposite is used to convert Australian dollars to euros. Use the graph to convert: a 5 Australian dollars to euros b 5 euros to Australian dollars Solution Read from the graph (when AUD = 5, EUR = ). Read from the graph (when EUR = 5, AUD = 5). a EUR b 5 AUD EUR Australian dollars to euros 5 AUD Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8/9/ 9: PM

19 Chapter 5 Interpreting linear relationships Eample Interpreting linear models Water is pumped into a partiall full tank. The graph gives the volume of water V (in litres) after t minutes. a How much water is in the tank at the start? b How much water is in the tank after minutes? c The tank holds 6 L. How long does it take to fill? d Find the equation of the straight line in terms of V and t. e Use the equation to calculate the volume of water in the tank after 7 minutes. f How man litres are pumped into the tank each minute? 6 8 Solution Read from the graph (when t =, V = ). a L Read from the graph (when t =, V = ). b L Read from the graph (when V = 6, t = ). c minutes Find the gradient b choosing two suitable points Rise such as (, ) and (, 6). d m = b = Run 5 Calculate the gradient (m) between these points 6 using the gradient formula. = 6 Determine the vertical intercept (). = 7 Substitute the gradient and -intercept into the = m + b gradient intercept form = m + b. V = t + 8 Use the appropriate variables (V for, t for ). 9 Substitute t = 7 into the equation. e V = t + Evaluate. Check the answer using the graph. = 7 + = L The rate at which water is pumped into the tank is the gradient of the graph. (m =) V f L/min Volume of water t Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8/9/ 9: PM

20 Preliminar Mathematics General Eercise 5E Water is pumped into a partiall full tank. The graph gives the volume of water V (in litres) after t minutes. a How much water is in the tank at the start? v Volume of water b How much water is in the tank after 5 minutes? c How much water is in the tank after 8 minutes? d The tank holds 5 L. How long does it take to fill? 5 e Use the graph to calculate the volume of water in the tank after 7 minutes. 5 The conversion graph opposite is used to convert Australian dollars to Chinese uan. Use the graph to convert: a 8 Australian dollars to uan b 5 Australian dollars to uan c uan to Australian dollars d 5 uan to Australian dollars e What is the gradient of the conversion graph? A post office charges according to the weight of a parcel. Use the step graph to determine the postal charges for the following parcels. a 5 g b 9 g c g d 8 g Cost ($) CNY Australian dollars to Chinese uan 6 8 t AUD Postal charges 6 8 Weight (g) Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8/9/ 9: PM

21 Chapter 5 Interpreting linear relationships 5 Development A new piece of equipment is purchased b a business for $. Its value is depreciated each month using the graph opposite. a What was the value of the equipment after months? v Value of equipment b What was the value of the equipment after one ear? c When does the line predict the equipment will 8 have no value? 6 d Find the equation of the straight line in terms of v and t. e Use the equation to predict the value of the equipment after months f B how much does the equipment depreciate Months in value each month? 5 The amount of mone transacted through ATMs has increased with the number of ATMs available. The graph below shows this increase. a What was the amount of mone transacted through ATMs when there were 5 machines? b How man ATM machines resulted in an amount of 75 billion? c Find the equation of the line in terms of amount of mone transacted, A, and the number of ATMs, N. d Use the equation to predict the amount of mone transacted when there were 5 machines. e Use the equation to predict how much mone will be transacted through ATM machines when there are machines. $ Amount of transactions A through ATMs Amount (billions of $) t 5 Number of machines (thousands) N Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 5 8/9/ 9: PM

22 6 Preliminar Mathematics General 6 A phone compan charges a monthl service fee, plus the cost of calls. The graph below gives the total monthl charge, C dollars, for making n calls. This includes the service fee. a How much is the monthl service fee? Total monthl b How much does the compan charge if ou make C charge calls a month? 7 c How man calls are made if the total monthl charge 6 is $? 5 d Find the equation of the line in terms of total monthl charge (C) and the number of calls (n) A compan charges the following parking fees: $ per hour for the first hours or part thereof, then $5 for the second hours or part thereof and $ for ever hour or part thereof after 6 hours. a Draw a step graph to illustrate the parking fees, with the Time (h) on the horizontal ais and Cost ($) on the vertical ais. b What is the cost to park for hours? Use the step graph. c Liam arrived in the parking area at. a.m. and left at. p.m. How much did he pa for parking? d Rub arrived in the parking area at 5.5 p.m. and left at.5 p.m. How much did he pa for parking? 8 Tomas converted Australian dollars to British pounds. a Draw a conversion graph with Australian dollars on the horizontal ais and British pounds on the vertical ais. b How man British pounds is Australian dollars? Use the conversion graph. c How man Australian dollars is British pounds? Use the conversion graph. d Find the gradient and vertical intercept for the conversion graph. e Write an equation that relates Australian dollars (AUD) to British pounds (GBP). n Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 6 8/9/ 9: PM

23 Chapter 5 Interpreting linear relationships 7 Chapter summar Interpreting linear relationships PowerPoint Graphing linear functions Construct a table of values with the independent variable as the first row and the dependent variable as the second row. Draw a number plane with the independent variable on the horizontal ais and the dependent variable as the vertical ais. Plot the points. Join the points to make a straight line. Gradient and intercept Gradient of a line is the slope of the line. Vertical rise Gradient (or m) = Horizontal run The intercept of a line is where the line cuts the aes. Gradient intercept formula Linear equation in the form = m + b. m Slope or gradient of the line. b -intercept. Sketching a straight line requires at least two points. When an equation is written in gradient intercept form, one point on the graph is immediatel available: the -intercept. A second point can be quickl calculated using the gradient. Simultaneous equations Draw a number plane. Graph both linear equations on the number plane. Read the point of intersection of the two straight lines. Linear functions as models Linear modelling occurs when a practical situation is described mathematicall using a linear function. Review Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 7 8/9/ 9: PM

24 8 Preliminar Mathematics General Review Sample HSC Objective-response questions An equation that compares the age of a fa machine (t) in ears to its current value (v) is v = -t + 5. What is the value of the fa machine after two ears? A 7 B 8 C 5 D What is the gradient of the line drawn opposite? A B C D Using the graph opposite what is the -intercept of this line? A B C D A straight line has the equation of = - +. What is the -intercept? A B C + D 5 A car is travelling at a constant speed. It travels 6 km in hours. This situation is described b the linear equation d = mt. What is the value of m? A.5 B C D 6 6 What is the point of intersection of the lines = + and = - +? A (, ) B (, ) C (, ) D (, ) 7 What is the equation of the line drawn opposite? A c = n B c = n + C c = n D c = 8n + 8 Using the graph opposite, what is the charge for months? A B 6 C D Monthl charge c 6 8 n Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 8 8/9/ 9: PM

25 Chapter 5 Interpreting linear relationships 9 Sample HSC Short-answer questions An internet access plan charges an ecess fee of $8 per GB. Data (d) 5 6 Cost (c) a Which is the dependent variable? b Which is the independent variable? c Draw a graph of data against cost. d Use the graph to find d if c is. One Australian dollar (AUD) was converted to.9 Japanese en (JPY). a Construct a table of values using,,, and as values for AUD and calculate the JPY using the above conversion. b Draw the graph of the AUD against JPY. What is the gradient of the line that joins these points? a (, 5) and (, 7) b (, ) and (, ) c (, ) and (, ) Draw a graph of these linear functions and find the gradient and -intercept. a = + b = c = - 5 Find the equations of the following line graphs. a b Review Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 9 8/9/ 9: PM

26 5 Preliminar Mathematics General Review Chapter 6 The graph summar opposite shows Earning the cost Mone of growing a rose plant and the income received from the sale of the roses. a Use the graph to determine the number of rose plants which need to be sold to break even. 8 Income b How much profit or loss is made when rose is sold? 6 c How much profit or loss is made when roses are sold? Costs d What are the initial costs? 7 A motor vehicle is purchased b a business for $. Its value is then depreciated each month using the graph opposite. a What was the value of the motor vehicle after months? b What was the value of the motor vehicle after one ear? c Find the equation of the straight line in terms of v and t. d Use the equation to predict the value of the motor vehicle after 6 months. e When does the line predict that the motor vehicle will have no value? f B how much does the motor vehicle depreciate in value each month? 8 The table below shows the speed v (in km/s) of a rocket at time t seconds. Time (t) 5 Speed (v) a Draw a number plane with time (t) on the horizontal ais and speed (v) on the vertical ais. b Plot the points from the table and draw a straight line through these points. c Etend the straight line to predict the rocket s speed when t = 6 seconds. Dollars ($) Dollars ($ ) 8 6 v Roses Motor vehicle 6 8 Time (months) t Uncorrected sample pages Cambridge Universit Press Powers Ph c5_p5-5.indd 5 8/9/ 9: PM