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?
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.
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.
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?
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.
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 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.
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 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.
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 |
Is this relationship linear?
Explain a method to check whether the relationship is linear, without having to plot the points.
The variable cost of running a business is \$150 an hour.
Find the total variable cost if the business operates for a total of 22 hours.
Express y, the total variable cost of the business, in terms of x, the total number of hours it has operated.
Find the number of hours the business has operated if it incurs total variable costs of \$5400.
Two siblings, Apollo and Artemis, leave home at different times. They are travelling to see their favourite sports team compete in the championship. The stadium is 30 \,\text{km} away and the match starts at 9:00 pm.
At what times did they both leave home?
Between which two times were Apollo and Artemis the same distance from home?
Who was travelling faster?
Did Apollo make it to the stadium on time?
This travel graph represents a train's journey between the airport and a nearby station:
How far did the train travel over the 10 minute period?
When was the train travelling the fastest?
Kenneth works for a delivery company. The following graph shows his distance from the warehouse throughout his shift:
How far was Kenneth from the warehouse at the start?
How far was he from the warehouse after 8 hours?
How many times did he visit the warehouse during this time period?
Calculate the total distance that he traveled between 1 and 4 hours.
When was Kenneth travelling the fastest?
The travel graph shows the first ten minutes of a student's journey home from school:
Describe their travel between the following times:
0 to 3 minutes
3 to 6 minutes
6 to 9 minutes
9 to 10 minutes
The Weber family travel 600\,\text{km} every year for their annual holidays.
Their distance from home on the trip this year is given in the following travel graph:
When did they stop for a break?
How far from their destination were they after 2 hours?
Identify two time periods when they were travelling at the same speed.
Castor and Pollux are working on a bushfire prevention team. Over the course of their 10-hour shift they move along a boundary and remove any flammable material they find.
The following travel graph shows their distance from the base throughout their shift:
How far from the base was Pollux when the shift began?
How far apart were Castor and Pollux after 2 hours?
When did Castor and Pollux meet up?
When was Pollux travelling the same speed as Castor's speed in his first three hours?
Paul is driving his child home from school. They travel 6\text{ km} in 10 minutes. After 5 minutes, the car slows down.
Construct a travel graph to represent this journey.
The manufacturer of a dishwasher wants to create a graph showing how one of their dishwasher models work. The stages of its operation are given below:
- The dishwasher quickly fills to half-way and performs a short wash.
- The dishwasher empties completely.
- The dishwasher completely fills up quickly and remains full over the course of the wash.
- The dishwasher empties completely.
Which of the following graphs match the water level inside the dishwasher throughout a cycle:
The graph shows the cost C \left(x\right) and the revenue R \left(x\right) from making and selling x units of a certain good:
State which line corresponds to the following functions:
C \left(x\right)
R \left(x\right)
How many units must be sold to reach the break-even point?
What is the revenue at the break-even point?
For each of the following cost revenue graphs:
State the coordinates of the break-even point.
State the fixed cost.
State which is the higher rate: the cost per unit or the revenue per unit.
The following graph shows the cost of producing drink bottles and the income from the sales of the drink bottles for a particular company:
For the line representing the cost of producing the drink bottles, find the:
y-intercept
gradient
Hence, state the equation of the line representing the cost of producing the drink bottles.
State the equation of the line representing the income from the sales of the drink bottles.
Find the quantity of bottles that need to be sold to break-even.
The following graph shows two lines that represent the revenue and cost from selling sandwiches at a local fair:
State the coordinates of the break-even point.
Find the equation of the line representing the revenue, y, from the sale of x sandwiches.
Find the equation of the line representing the cost, y, of making x sandwiches.
Would the stand make a profit or a loss if they sold 70 sandwiches?
The cost of manufacturing toys, C, is related to the number of toys produced, n, by the formula C = 400 + 2 n. The revenue, R, made from selling n toys is given by R = 4 n.
Sketch the graph of the cost and the revenue on the same number plane.
How many toys need to be produced for the revenue to equal the cost?
Explain the meaning of the y-coordinate of the point of intersection.
Consider the following phone plans:
GO SMALL plan: This plan has a \$20 monthly base charge and charges 90\text{c} per minute for all calls.
GO MEDIUM plan: This plan has a \$26 monthly base charge and then charges 70 \text{c} per minute for all calls.
Complete the table of values for various total monthly call times for the two plans:
| \text{Call time} \\ \text{(in minutes)} | \text{Total cost for} \\ \text{GO SMALL plan } (S) | \text{Total cost for} \\ \text{GO MEDIUM plan } (M) |
|---|---|---|
| 20 | ||
| 30 | ||
| 40 | ||
| 50 |
Sketch a graph for the two plans on the same set of axes.
Hence, find how many minutes of calls would need to be made so that the monthly bill costs the same on both plans.
The cost of manufacturing car parts \left(C\right) is related to the number of car parts produced \left(n\right) by the formula C = 6000 + 4 n.
The revenue \left(R\right) made from selling n car parts is R = 8 n - 2000.
Complete the following table:
| Number of car parts | Cost (in dollars) | Revenue (in dollars) |
|---|---|---|
| 1000 | ||
| 2000 | ||
| 3000 | ||
| 4000 |
Sketch the graph of the cost and the revenue functions on the same set of axes.
Find the number of car parts that need to be produced for revenue to equal cost.
Explain the meaning of the y-coordinate of the break-even point.
The two equations y = 3 x + 35 and y = 4 x represent Laura’s living expenses and income from work respectively.
Sketch both equations on the same number plane.
Find the point of intersection of the two equations.
Explain the meaning of the point of intersection of the two lines.
The two equations y = 4 x + 400 and y = 6 x represent a company's revenue and expenditure respectively.
Sketch both equations on the same number plane.
Find the point of intersection of the two equations.
Explain the meaning of the point of intersection of the two lines.
An electronics manufacturer has found the cost of creating circuits is represented by the equation C = 80 + 2 x, and the income received from selling them is represented by the equation I = 2.8 x, where x represents the number of circuits.
Complete the table of values for both the cost and income functions:
| x | 0 | 50 | 100 | 150 | 200 | 250 | 300 |
|---|---|---|---|---|---|---|---|
| \text{Cost in dollars } (C) | |||||||
| \text{Income in dollars } (I) |
Graph the cost and income functions on the same set of axes.
Hence, find the point that satisfies both equations simultaneously.
Interpret the meaning of the point of intersection in this context.
The cost for a printing company to make a large banner is \$500 per banner plus a fixed setup cost of \$3500. The banners will sell for \$650 each.
Write an expression to represent the cost of manufacturing x banners.
Write an expression to represent the revenue generated from the sale of x banners.
Find the x-coordinate of the break-even point.
Explain the meaning of the x-coordinate of the break-even point.
The cost for a furniture manufacturer to make an armchair is \$600 per armchair plus a fixed setup cost of \$8500. The armchairs will sell for \$850 each.
Write an expression to represent the cost of manufacturing x armchairs.
Write an expression to represent the revenue generated from the sale of x armchairs.
Find the break-even point.
Explain the meaning of the break-even point.
The cost for a furniture manufacturer to make a dining table is \$450 per dining table plus a fixed setup cost of \$6000. The dining tables will sell for \$700 each.
Calculate the cost of manufacturing 42 dining tables.
Write an expression to represent the cost of manufacturing x dining tables.
Find the revenue that is generated by the sale of 42 dining tables.
Write an expression to represent the revenue generated from the sale of x dining tables.
If n is the whole number of dining tables that need to be sold for the company to break-even, find n.
The monthly cost, C \left(x\right), revenue, R \left(x\right) and profit, P \left(x\right) functions for a cleaning company are given below, where x represents the number of clients in a month:
C \left(x\right) = 21 x + 5760; \, R \left(x\right) = 57 x; \, \, P \left(x\right) = 36 x - 5760
What is the company's fixed cost, even with no clients?
How much per month does the company charge each client?
Find the value of x, the number of services provided that allows them to break-even.
For each pair of cost, C \left( x \right), and revenue functions, R \left( x \right), find the number of units, x, that must be sold to break-even:
C \left( x \right) = 20 x + 8100 \\ \ R \left( x \right) = 32 x
C \left( x \right) = 0.3 x + 1275 \\ \ R \left( x \right) = 2 x
C \left( x \right) = 0.4 x + 2015 \\ R \left( x \right) = 3 x