Consider the graph which shows the cost of a consultation with a medical specialist for a student or an adult, according to the length of the consultation:
Find the cost for an adult consultation of 9 minutes.
Find the cost for a student consultation of 9 minutes.
Determine the percentage discount for a student consultation.
The conversion rate between the Australian dollar and the Euro on a particular date was \\1 AUD =0.7 EUR.
Graph the relationship between the Australian Dollar and the Euro, with AUD on the horizontal axis and EUR on the vertical axis of a number plane.
Calculate the gradient of the line.
Using the graph, convert 5 AUD into EUR.
Using the graph, convert 10.5 EUR into AUD.
Mario wants to determine which of two slow-release pain medications is more rapidly absorbed by the body.
Consider the graph and table below, which show the amount of medication in the bloodstream for the liquid and capsule form of the medication:
Liquid form
Capsule form
| \text{Time (mins)}, t | \text{Amount in} \\ \text{blood (mgs)}, A |
|---|---|
| 4 | 24.6 |
| 7 | 42.3 |
| 10 | 60 |
| 13 | 77.7 |
Use the graph to find the rate, in \text{mg} per minute, in which the liquid form is absorbed.
Use the table to find the rate, in \text{mg} per minute, in which the capsule form is absorbed.
In which form is the medication absorbed more rapidly?
The graph shows the temperature of a room after the heater has been turned on.
State the gradient of the line.
State the y-intercept.
Write an equation to represent the temperature of the room, y, as a function of time, t.
Explain the meaning of the gradient in this context.
Explain the meaning of the y-intercept in this context.
Find the temperature of the room after the heater has been turned on for 40 minutes.
The graph shows the conversion between temperatures in Celsius and Fahrenheit:
Use the graph to convert 10 \degree \text{C} into Fahrenheit.
0 \degree \text{C} is 32 \degree \text{F}. Hence, for every 1 \degree \text{C} increase, by how much does the Fahrenheit temperature increase?
Would 80 \degree \text{F} be above or below normal body temperature (approximately 37 \degree \text{C})?
Write the rule for conversion between Celsius \left(\text{C}\right) and Fahrenheit \left(\text{F}\right).
Convert 35 \degree \text{C} into Fahrenheit.
The graph shows the amount of Euros that can be bought with Australian Dollars on a particular date:
How many Euros can \$20 AUD buy?
How much Australian currency is required to buy 6 Euros?
Calculate the number of Euros that \\ \$1 AUD buys.
Consider the following graph which relates the two units, Indonesian rupiah (IDR) in hundreds of thousands, and Australian dollars (AUD):
If Holly goes on holiday to Bali and spends 50\,000 IDR per day on food for 4 days, how much in AUD would this be?
How many rupiahs would \$50 AUD buy?
State the exchange rate between AUD and IDR.
Valentina left for a road trip at midday. The graph shows the total distance travelled (in kilometres) t hours after midday.
Let the horizontal axis represent the time in hours and the vertical axis represent the distance travelled in kilometers.
State the gradient of the line.
What does the gradient of the line represent in this context?
The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in it's side.
State the gradient of the line.
Find the y-intercept.
Write an equation to represent the amount of water remaining in the bucket, y, as a function of time, x.
Explain the meaning of the gradient in this context.
What does the y-intercept represent in this context?
Find the amount of water remaining in the bucket after 54 minutes.
There are 20 \text{ L} of water in a rainwater tank. It rains for a period of 24 hours and during this time the tank fills up at a rate of 8 \text{ L/h}. Complete the table of values:
| \text{Number of hours passed }(x) | 0 | 4 | 6 | 7 | 9 | 11 | 12 |
|---|---|---|---|---|---|---|---|
| \text{Amount of water in tank }(y) |
The cost of a taxi ride C is given by C = 2.50 t + 3 where t is the duration of the trip in minutes.
Calculate the cost of a 6-minute trip.
Calculate the cost of a 7-minute trip.
Complete the table of values:
| \text{Time in minutes }(t) | 6 | 7 | 8 | 9 | 11 | 16 |
|---|---|---|---|---|---|---|
| \text{Cost in dollars }(C) |
A racing car starts the race with 250 litres of fuel. From there, it uses fuel at a rate of 5 litres per minute.
Complete the table of values:
| \text{Number of minutes passed, }x | 0 | 5 | 10 | 15 | 20 | 50 |
|---|---|---|---|---|---|---|
| \text{Amount of fuel left in tank, }y |
Determine an algebraic rule linking the number of minutes passed, x, and the amount of fuel left in the tank, y.
Describe how the amount of fuel in the car is changing over time.
A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table below shows the depth of the diver over 4 minutes:
| \text{Number of minutes passed, }x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Depth of diver in metres, }y | 0 | 1.4 | 2.8 | 4.2 | 5.6 |
Calculate the increase in depth each minute.
Write a linear equation for the relationship between the number of minutes passed, x, and the depth, y, of the diver.
Calculate the depth of the diver after 6 minutes.
Calculate how long the diver takes to reach 12.6 metres beneath the surface.
After Mae starts running, her heart rate in beats per minute increases at a constant rate as shown in the following table:
| \text{Number of minutes passed, }x | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|---|
| \text{Heart rate, }y | 49 | 55 | 61 | 67 | 73 | 79 |
Determine Mae's heart rate after 12 minutes.
Calculate the change in heart rate per minute.
Write an equation that describes the relationship between the number of minutes passed, x, and Mae’s heart rate, y.
The amount of medication M (in milligrams) in a patient’s body gradually decreases over time t (in hours) according to the equation M = 1050 - 15 t.
After 61 hours, how many milligrams of medication are left in the body?
Calculate the number of hours it will take for the medication to be completely removed from the body.
Consider the pattern of blue boxes below:
Complete the table of values below:
| \text{Number of columns }(c) | 1 | 2 | 3 | 5 | 10 | 20 |
|---|---|---|---|---|---|---|
| \text{Number of blue boxes }(b) |
Write a formula that describes the relationship between the number of blue boxes \left(b\right) and the number of columns \left(c\right).
Calculate the number of blue boxes if this pattern were to continue for 38 columns.
If there were 45 blue boxes, how many columns would there be?
A carpenter charges a callout fee of \$150 plus \$45 per hour.
Write a linear equation to represent the total amount charged, y, by the carpenter as a function of the number of hours worked, x.
State the gradient of the linear function.
Explain the meaning of the gradient in this context.
State the value of the y-intercept.
Explain the meaning of the y-intercept in this context.
Find the total amount charged by the carpenter for 6 hours of work.
Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol in dollars:
| \text{Number of litres }(x) | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| \text{Cost of petrol }(y) | 0 | 16.40 | 32.80 | 49.20 | 65.60 |
Find the cost of petrol per litre.
Write an equation linking the number of litres of petrol pumped \left(x\right) and the cost of the petrol \left(y\right).
Explain the meaning of the gradient in this context.
Calculate the cost of 47 \text{ L} petrol.
The table shows the linear relationship between the number of plastic chairs manufactured, x, and the total manufacturing cost, y:
| Number of plastic chairs | 5 | 10 | 15 |
|---|---|---|---|
| Cost (dollars) | 135 | 185 | 235 |
State the gradient of the line.
Form an equation relating x and y.
Find the y-intercept.
Explain the meaning of the y-intercept in this context.
Explain the meaning of the gradient of the function in this context.
Find the total cost of manufacturing 25 plastic chairs.
The number of calories burned by the average person while dancing is modelled by the equation C = 8 m, where m is the number of minutes.
Sketch the graph of this equation to show the calories burnt after each 15-minute interval.
The number of university students studying computer science in a particular country is modelled by the equation S = 4 t + 12, where t is the number of years since 2000 and S is the number of students in thousands.
Sketch the graph of this equation to show the number of computer science students at the end of each 4-year period.
Let the height of a candle be y \text{ cm} . If the candle is lit, the height decreases according to the equation y = - 2 t + 8, where t is the elapsed time in minutes.
Complete the table of values:
| \text{Time }(t \text{ min}) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \text{Height of candle }(y \text{ cm}) |
Sketch the graph of y = - 2 t + 8.
The height of candle and the time elapsed must be non-negative. Hence, state the only posible values of t.
A dam used to supply water to the neighboring town had the following data recorded for its volume over a number of months:
| \text{Month }(M) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Volume in billions of litres } (V) | 112 | 106 | 110 | 80 |
Plot the points on the number plane.
Is this relationship linear?
Let the volume of water in a tank be V \text{ L} . If the tank is initially empty, then the amount of water in the tank is given by V = 5 t where t is the elapsed time in minutes and 5 is the rate that the tank fills in \text{L/min}.
Complete the table of values below:
| \text{Time }(t \text{ min}) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \text{Volume of water }(V \text{ cm}) |
Sketch the graph of V = 5 t.
The volume of the tank is 15 \text{ L} and the time elapsed is non-negative. Hence, state the only posible values of t.
The amount of medication in a patient’s body \left(M \text{ mg}\right) gradually decreases over time \left(t \text{ hours}\right) according to the equation M = 12 - 2 t.
Initially, at t = 0, state the medication present in the body, M.
Calculate how many hours, t, will it take for the medication to be completely removed from the body.
Hence, graph the line M = 12 - 2 t.
Beth’s income is based solely on the number of hours she works, and she is paid a fixed hourly wage. She earns \$750 for working 30 hours. Let y represent Beth’s income after working x hours.
Sketch a graph that displays her income against her hours worked.
Form an equation relating x and y.
Calculate Beth's income when she works 25 hours.
Calculate the number of hours that Beth must work to earn \$125.
The cost, y, for a business to operate, can be expressed in terms of x, the total number of hours it has operated for. The cost is \$120 an hour.
Sketch a graph that displays the cost against time.
State the gradient of the line.
Form an equation relating x and y.
Find the total cost for the business to operate for 28 hours.
Find the number of hours that the business needs to operate to incur a total cost of \$3840.
Consider the following table that shows the temperature of a metal plate, in \degreeC, after an amount of time, measured in minutes:
| \text{Time }(x) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Temperature }(y) | 10 | 15 | 20 | 25 | 30 |
Graph the linear relationship represented in the table.
By how much is the temperature increasing each minute?
Find the initial temperature.
Form an equation relating x and y.
Charlie is on a holiday and has been spending money at a rate of \$50 per day. After 13 days of the holiday, he has \$450 remaining as shown in the following table:
| \text{Day, }d | 4 | 9 | 10 | 13 |
|---|---|---|---|---|
| \text{Money remaining, }t | \$900 | \$650 | \$600 | \$450 |
Plot the given data on a number plane and connect the points with a line.
How much money did Charlie have at the beginning of the trip?
How much money did Charlie have after the 18th day?
At the end of which day did Charlie have \$850 remaining?
At the end of which day did Charlie have \$100 remaining?
Determine the rule that connects the amount of money remaining, t, to the days away on holiday, d.
Justin is looking into the details of his mobile phone plan. He knows the costs for several call lengths as shown in the table:
| \text{Length of call in minutes } (t) | 2 | 6 | 10 | 14 |
|---|---|---|---|---|
| \text{Cost } (C) | \$1.00 | \$2.20 | \$3.40 | \$4.60 |
Plot the given data on a number plane and connect the points with a line.
How much will it cost to make an 8-minute call?
Find the length of a call that costs \$1.60.
Determine the rule that connects the cost of a call, C, to the length of the call, t.
To measure the effectiveness of a new train timetable, the average waiting time after a train was due at Southern Cross Station was recorded for 4 months in the following table:
| \text{Month} | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Average waiting time (seconds)} | 30 | 50 | 70 | 90 |
Let x be the number of months and y be the average waiting time in seconds.
Plot the given data on a number plane and connect the points with a line.
What will the average waiting time be in 6 months?
What will the average waiting time be in 1 year?
After how many months will the average waiting time be 170 seconds?
After how many months will the average waiting time be 230 seconds?
Determine the rule that connects the average waiting time in seconds, y, to the number of months, x, that have passed.
A ball is rolled down a slope. The table below shows the velocity of the ball after a given number of seconds:
| \text{Time in seconds, }t | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Velocity, }V | 12 | 13.3 | 14.6 | 15.9 | 17.2 | 18.5 |
Determine the rule that connects the velocity, V, to the time in seconds, t.
Use your CAS calculator to graph the line that represents the relationship between velocity and time.
Explain the meaning of the gradient of the line in this context.
Explain the meaning of the vertical intercept of the line in this context.
Find the velocity of the ball after 19 seconds, rounded to one decimal place.
Luke purchases 30\text{ L} of mixed leaded and unleaded petrol solution, which costs him a total of \$103.20. The price of leaded petrol is \$3.54\text{/L}, while the price of unleaded petrol is \$2.94\text{/L}.
Let x and y be the number of litres of leaded and unleaded petrol that make up the solution respectively.
Use the fact that Luke purchased a total of 30\text{ L} of petrol to write an equation involving x and y.
Use the fact that Luke purchased a total of \$103.20 worth of petrol to set up a second equation involving x and y.
Use the graphing function of your CAS calculator to sketch the graphs of the two equations on the same number plane.
Hence determine the amount of leaded petrol that was used in the solution.
Determine the amount of unleaded petrol that was used in the solution.
The number of fish in a river is approximated over a five year period. The results are shown in the following table.
| \text{Time in years }(t) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Number of fish }(F) | 4800 | 4600 | 4400 | 4200 | 4000 | 3800 |
Sketch a graph that corresponds to this information.
Calculate the gradient of the line.
What does the gradient represent in this context?
State the value of F when the line crosses the vertical axis.
Determine an equation for the line, using the given values.
Hence, determine the number of fish remaining in the river after 13 years.
Find the number of years, \left(t\right), until 2000 fish remain in the river.