The graph shows the conversion between temperatures in Celsius and Fahrenheit. Note that 0 \degree \text{C} is 32 \degree \text{F}.
Use the graph to convert 10 \degree \text{C} into \degree \text{F}.
Calculate the gradient of the line as a decimal number.
For every increase by 1 \degree \text{C}, by how much does the Fahrenheit temperature increase?
The graph shows the conversion between Country A and Country B's currency:
Use the graph to convert 8 of currency A to currency B.
Use the graph to convert 2 of currency B to currency A.
Calculate the gradient of the line.
Hence state the exchange rate to convert currency B to currency A.
Consider the conversions graph for miles and kilometres:
If a car is driving 32\text{ km/hour} in a school zone, at what speed are they travelling in miles/hour?
Calculate the conversion rate to change \text{km/hour} to \text{miles/h}.
Calculate the gradient of the line.
A road sign states the speed limit to be 128 kilometres per hour. What is this speed limit in miles per hour?
What does the gradient represent in this situation?
The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in it's side.
Find the gradient of the line.
State the y-intercept.
Hence write an equation to represent the amount of water remaining in the bucket, y, as a function of time, x.
Find the amount of water remaining in the bucket after 54 minutes.
Explain the meaning of the gradient in this context.
What does the y-intercept represent in this context?
Valentina left for a road trip at midday and travels at a constant average speed. 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 kilometres.
State the gradient of the line.
What does the gradient of the line represent in this context?
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 10 minutes.
Find the cost for a student consultation of 10 minutes.
Calculate the discount for the hourly rate of a student consultation.
Determine the percentage discount for the hourly rate of a student consultation.
The graph shows the temperature of a room (in degrees Celcius) against the time since the heater has been turned on (in minutes).
Find the gradient of the function.
State the y-intercept.
Hence write an equation to represent the temperature of the room, y, as a function of time, x.
Find the temperature of the room after the heater has been turned on for 30 minutes.
What does the gradient represent in this context?
What does the y-intercept represent in this context?
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.
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.
Mohamad is taking his new Subaru out for a drive. He had only driven 50 kilometres in it before and is now driving it down the highway at 75\text{ km/h} .
Write an equation to represent the total distance, y, that Mohamad had driven in his Subaru as a function of the number of hours, x.
State the gradient of the function.
Describe what the gradient of the line represents in context.
Find of the y-intercept.
Describe what the y-intercept represents in context.
Find the total distance Mohamad will have driven in his Subaru if his current drive begins at 5:10 pm and finishes at 7:25 pm.
Mario is running a 100 \text{ km} ultramarathon at an average speed of 9 \text{ km/h}.
Write an equation to represent the distance Mario has left to run, y, as a function of the number of hours since the start, x.
State the gradient of the function.
Describe what the gradient of the line represents in context.
Find the y-intercept.
Describe what the y-intercept represents in context.
Find the distance Mario will have left to run after 4.5 hours.
A particular restaurant has a fixed weekly cost of \$1300 and receives an average of \$16 from each customer.
Write an equation to represent the net profit, y, of the restaurant for the week as a function of the number of customers, x.
Find the gradient of the function.
Describe what the gradient of the line represents in context.
Find the y-intercept.
Describe what the y-intercept represents in context.
Find the restaurant's net profit if it has 310 customers for the week.
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.
In a study, scientists found that the more someone sleeps, the quicker their reaction time. The table below displays the findings:
| \text{Number of hours of sleep } (x) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Reaction time in seconds } (y) | 6 | 5.8 | 5.6 | 5.4 | 5.2 | 5 |
How much does the reaction time decrease for each extra hour of sleep?
Write an algebraic equation relating the number of hours of sleep (x) and the reaction time (y).
Calculate the reaction time for someone who has slept 4.5 hours.
Calculate the number of hours someone sleeps if they have a reaction time of 5.5 seconds.
A diver starts at the surface of the water and starts to descend below the surface at a constant rate. The table shows the depth of the diver over 5 minutes.
| Number of minutes passed | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Depth of diver in metres | 0 | 0.8 | 1.6 | 2.4 | 3.2 |
Graph the linear relationship represented in the table.
What is the increase in depth each minute?
State the gradient of the line.
Calculate the depth of the diver after 24 minutes.
Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol in dollars:
| \text{Number of litres } (x\text{)} | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| \text{Cost of petrol } (y\text{)} | 0 | 12.00 | 24.00 | 36.00 | 48.00 |
Graph the linear relationship represented in the table.
How much does petrol cost per litre?
How much would 75 litres of petrol cost at this unit price?
Consider the following table that shows the temperature of a metal plate, in \degree\text{C}, 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.
Hence form an equation relating x and y.
Find the temperature of the plate after 12 minutes.
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.
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.
State how much medication, M, was present in the body inititally, at t = 0.
Find the value of t when M=0.
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.
Find the gradient of the line.
Explain the meaning of the gradient in this context.
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.
A ball is rolled down a slope. The table below shows the velocity (V) of the ball after a given number of seconds (t):
| \text{Time in seconds } (t) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Velocity in m/s } (V) | 12 | 13.3 | 14.6 | 15.9 | 17.2 | 18.5 |
Plot the graph of the ball's velocity against time on a coordinate plane.
Calculate the gradient of the line.
What does the gradient represent in this context?
State the vertical intercept of the line.
What does the vertical intercept represent in this context?
Write an algebraic equation for the line, expressing V in terms of t.
Hence determine the velocity of the ball after 19 seconds. Round your answer to one decimal place.
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.
A racing car starts the race with 140 litres of fuel. From there, it uses fuel at a rate of 2 litres per minute. Complete the table of values:
| \text{Number of minutes passed} \,(x) | 0 | 5 | 10 | 15 | 20 | 70 |
|---|---|---|---|---|---|---|
| \text{Amount of fuel left}\, (y) |
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. Complete the table of values:
| \text{Time in minutes }(t) | 6 | 7 | 8 | 9 | 11 | 16 |
|---|---|---|---|---|---|---|
| \text{Cost in dollars }(C) |
The cost of a taxi ride is given by C = 5.5 t + 3, where t is the duration of the trip in minutes.
Calculate the cost of an 11-minute trip.
For every extra minute the trip takes, how much more will the trip cost?
What could the constant value of 3 represent in context?
A baseball is thrown vertically upward by a baseball player when he is standing on the ground, and the velocity of the baseball V (in metres per second) after T seconds is given by V = 120 - 32 T.
Complete the table of values:
| \text{Time} | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Vertical Velocity} |
State the gradient of the linear function.
Explain the negative value of V when T = 4.
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 linear function.
Form an equation relating x and y.
Find the y-intercept.
Find the total cost of manufacturing 25 plastic chairs.
Explain the meaning of the y-intercept in this context.
Explain the meaning of the gradient of the function in this context.
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.
After Sally starts running, her heartbeat increases at a constant rate:
Complete the table:
| \text{Number of minutes passed } (x\text{)} | 0 | 2 | 4 | 6 | 8 | 10 | 11 |
|---|---|---|---|---|---|---|---|
| \text{Heart rate } (y\text{)} | 75 | 81 | 87 | 93 | 99 | 105 |
State the gradient of the line.
By how much is her heartbeat increasing each minute?
Explain the meaning of the y-intercept in this context.
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 |
Find the increase in heart rate for every 2 minutes.
Determine Mae's heart rate after 12 minutes.
Find the increase 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 in the form y=mx+b.
Consider the points in the table, where the time (x) is measured in minutes:
| \text{Time } (x\text{)} | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Temperature } (y\text{)} | 8 | 13 | 18 | 23 | 28 |
By how much is the temperature increasing each minute?
Find the initial temperature at time 0.
Find the algebraic rule between x and y.
Find the temperature after 17 minutes.
A racing car starts the race with 150 \text{ L} of fuel. From there, it uses fuel at a rate of 5\text{ L} per minute.
Complete the following table of values:
| \text{Number of minutes passed } (x) | 0 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| \text{Amount of fuel left in the tank } (y) |
Write an algebraic relationship linking the number of minutes passed \left(x\right) and the amount of fuel left in the tank \left(y\right).
How many minutes will it take for the car to run out of fuel?
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.
Explain the meaning of the gradient in this context.
Consider the pattern for blue boxes below:
Complete the table:
| \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 (b) and the number of columns (c).
State the number of blue boxes, b, required for:
38 columns
92 columns
State the number of columns, c, that would contain:
45 blue boxes
51 blue boxes
A dam used to supply water to the neighbouring 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 |
Is this relationship linear?
Explain a method to check whether the relationship is linear, without having to plot the points.
James recorded his savings (in dollars) over a few months in the graph given.
Complete the table:
| \text{Months} | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Savings } \left(\$\right) |
Is James correct if he estimates that he will have exactly \$60 in his savings by month 5?
The graph shows the relationship between the number of cartons and the number of eggs in them.
Complete the table:
| \text{Cartons} | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Eggs} |
The graph shows the relationship between water temperatures and surface air temperatures:
Complete the table of values:
| \text{Water Temperature } \left(\degree \text{C} \right) | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| \text{Surface Air Temperature } \left(\degree \text{C} \right) |
Write an algebraic equation representing the relationship between the water temperature (x) and the surface air temperature (y).
Find the surface air temperature when the water temperature is 14 \degree \text{C}.
Find the water temperature when the surface air temperature is 23 \degree \text{C}.
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.
Calculate the time when y=0.
The height of candle and the time elapsed must be non-negative. Hence, state the only posible values of t.
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.
A car travels at an average speed of 75\text{ km/h}.
Complete the table of values for D = 75 t, where D is the distance travelled in kilometres and t is the time taken in hours:
| t | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| D |
How far will the car travel in 9 hours?
Sketch the graph of D = 75 t on a coordinate plane.
State the gradient of the line.
How long would it take for the car to travel a distance of D=675\text{ km} at the given speed?
It starts raining and an empty rainwater tank fills up at a constant rate of 2 litres per hour. By midnight, there are 20 litres of water in the rainwater tank. As it rains, the tank continues to fill up at this rate.
Complete the table of values:
| \text{Number of hours passed since midnight } (x) | 0 | 1 | 2 | 3 | 4 | 4.5 | 10 |
|---|---|---|---|---|---|---|---|
| \text{Amount of water in tank } (y) | 20 | 22 | 24 |
Sketch the graph depicting the situation on a coordinate plane.
Write an algebraic relationship linking the number of hours passed since midnight (x) and the amount of water in the tank (y) in the form y=mx+b.
Calculate the value of x when the tank is empty, when y=0.