Deborah left for a road trip at midday. The following graph shows the total distance travelled (in kilometres), t hours after midday:
How far has the car travelled after 8 hours?
Find the slope.
Describe what the slope of the line represents in context.
The plotted points show the relationship between water temperatures, x, and surface air temperatures, y:
Complete the table of values:
| x | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y |
Continue the pattern in the table to determine the surface air temperature when the water temperature is 4 \degree \text{C}.
What would the surface air temperature be when the water temperature is 14 \degree \text{C}?
A husband and wife exercise each day for 20 minutes before dinner. The wife walks briskly, while the man runs. The distance each of them travel is shown on the graph:
Find the difference in distance that each of them covers after 20 minutes.
Find the distance the wife covers each minute.
Find the distance the husband covers each minute.
How long would it take the wife to walk the same distance that her husband runs in 6 minutes?
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.
David decides to start his own yoga class. The cost and revenue functions of running the class have been graphed:
Calculate the amount of revenue David receives for each student.
Determine the number of students that must attend the class so that David can cover his costs.
Calculate the profit David makes if there are 8 students in his class.
The graph shows the amount of water remaining in a bucket that was initially full before a hole was drilled in it's side:
Find the slope.
State the y-intercept.
Find the equation for the amount of water remaining in the bucket, y, as a function of time, x.
Describe what the slope of the line represents in context.
Describe what the y-intercept represents in context.
Find the amount of water remaining in the bucket after 54 minutes.
The graph shows the temperature of a room after the heater has been turned on.
Calculate the slope of the function.
State the y-intercept.
Write an equation to represent the temperature of the room, y, as a function of time, x.
Describe what the slope of the line represents in context.
Describe what the y-intercept represents in context.
Find the temperature of the room after the heater has been turned on for 40 minutes.
The graph shows the amont of money Iain earns from his job as a librarian, given the number of hours he worked that week.
How much would Iain be paid if he worked 4 hours?
How long does Iain need to work to get paid \$120?
How much does Iain earn per hour?
Write down the equation of the line that represents the amount of money earned, y, in terms of the number of hours worked, x.
Hence, using the equation, find the amount he would be paid for 32 hours.
How many hours does he have to work to earn \$260?
The number of calories, C, 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 = 12 + 4 t , 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.
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.
Sketch the graph that depicts her income (y), against her hours worked (x).
Calculate the amount Beth earns each hour.
Write an equation relating x and y.
Determine Beth's income when she works 25 hours.
How many hours must Beth work if she wants to earn \$125?
The conversion rate between the Australian dollar (x) and the Euro (y) is approximately:
1\text{ AUD} =0.7\text{ EUR}
Sketch the graph that depicts the relationship between the Australian Dollar and the Euro.
Calculate the slope of the line.
Use the graph to convert 5\text{ AUD} into EUR.
Use the graph to convert 10.5\text{ EUR} into AUD.
Rosey earns an hourly wage of \$17.40 an hour.
Sketch the graph of her total wages against the number of hours worked.
State the slope of the line.
Express y, Rosey's total wages, in terms of x, the number of hours worked.
Find her total wages if she works a total of 25 hours.
Find the number of hours she must work to earn \$696.00.
A mobile phone carrier charges 1.1\, cents per second for each call, with no connection fee.
Sketch the graph that depicts the cost of a call in dollars (y) against the call length (x) in seconds.
Calculate the slope of the line.
Express y in terms of x.
Find the cost of a call that lasts 60 seconds.
Find the length of the call that costs \$1.32.
The variable cost of running a business is \$110 an hour.
Sketch the graph that depicts the total variable cost (y) against time (t) in hours.
Calculate the slope of the line.
Express y in terms of t.
Find the total variable cost if the business operates for a total of 26 hours.
Find the number of hours the business has operated for if it incurs total variable costs of \$3960.
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 slope of the line.
If the destination is 675\text{ km} ahead, how long would it take for the car to reach the destination at the given speed?
After Mae starts running, her heartbeat increases at a constant rate.
Complete 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 |
What is the unit change in y for the above table?
Write an equation that describes the relationship between the number of minutes passed (x) and Mae’s heartbeat (y).
In the equation, y = 49 + 3 x , what does the 3 represent in context?
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?
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) |
Plot 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).
Determine the y-intercept of the line.
At what time prior to midnight was the tank empty?
The table shows the linear relationship between the length of a mobile phone call and the cost of the call:
| \text{Length of call (mins)}, x | 1 | 2 | 3 |
|---|---|---|---|
| \text{Cost } (\$), y | 7.6 | 14.4 | 21.2 |
Write an equation to represent the cost of a call, y, as a function of the length of the call, x.
State the slope of the function.
Describe what the slope represents in context.
State the y-intercept.
Describe what the y-intercept could represent in context.
Find the cost of a 6-minute call.
The table shows the linear relationship between the number of plastic chairs manufactured and the total manufacturing cost:
| \text{No. of plastic chairs}, x | 2 | 4 | 7 |
|---|---|---|---|
| \text{Cost } (\$), y | 135 | 185 | 260 |
Write an equation to represent the total manufacturing cost, y, as a function of the number of plastic chairs manufactured, x.
State the slope of the function.
Describe what the slope of the function represents in context.
State the y-intercept.
Describe what the y-intercept could represent in context.
Find the total cost of manufacturing 13 plastic chairs.
The table shows the water level of a well that is being emptied at a constant rate with a pump:
| Time (minutes) | 2 | 5 | 8 |
|---|---|---|---|
| Water level (metres) | 26.8 | 25 | 23.2 |
Write an equation to represent the water level, y, as a function of the minutes passed, x.
Calculate the slope of the function.
Describe what the slope of the function represents in context.
State the y-intercept.
Describe what the y-intercept represents in context.
Calculate the water level be after 15 minutes.
The table shows Peter's earnings from sewing shirts.
| \text{Shirts sewed } (x) | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| \text{Earnings } (y) | 4 | 8 | 12 | 16 | 20 |
Sketch the graph of his earnings against the number of shirts he sews.
Calculate the slope of the line.
Describe what the slope represents in context.
State the y-intercept of the line.
Describe what the y-intercept represents in context.
Calculate his total earnings if he produces a total of 14 shirts.
How many shirts will he have to produce in order to earn \$28?
A ball is rolled down a slope. The table below shows the velocity of the ball after a given number of seconds:
| \text{Time (seconds), }t | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Velocity(m/s), }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.
Describe the meaning of the slope of the line in this context.
Describe 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.
Kerry currently pays \$50 a month for her internet service. She is planning to switch to a fibre optic cable service.
Complete the table of values for the total cost of the current internet service:
Write an equation for the total cost, T, of Kerry's current internet service over a period of n months.
| n | 1 | 6 | 12 | 18 | 24 |
|---|---|---|---|---|---|
| T \, (\$) |
For the fibre optic cable service, Kerry must pay a one-off amount of \$1200 for the installation costs and then a monthly fee of \$25.
Complete the table of values for the total cost of the fibre optic cable service:
Write an equation for the total cost T of Kerry's new internet service over n months.
| n | 1 | 6 | 12 | 18 | 24 |
|---|---|---|---|---|---|
| T \,(\$) |
Sketch the pair of lines that represent the costs of the two internet services on a number plane.
Determine how many months it will take for Kerry to break even on her new internet service.
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 5 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.
The number of fish in a river is approximated over a five year period. The results are shown in the table below:
| \text{Time in years }(t) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Number of fish }(F) | 4800 | 4600 | 4400 | 4200 | 4000 | 3800 |
Sketch the graph of the relationship on a coordinate plane.
Calculate the slope of the line.
Describe what the slope of the line represents in context.
State the value of F when the line crosses the vertical axis.
Write an algebraic equation for the line relating t and F.
Hence determine the number of fish remaining in the river after 13 years.
Determine the number of years it takes for there to be 2000 fish remaining in the river.
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.
The cost of a taxi rideis given by C = 3 + 5.5 t, 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?
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?
How many hours will it take for the medication to be completely removed from the body?
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 slope of the linear function.
Describe what the slope of the line represents in context.
State the value of the y-intercept.
State 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 miles in it before and is now driving it down the highway at 75\text{ mi/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 slope of the function.
Describe what the slope 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 slope of the function.
Describe what the slope 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 slope of the function.
Describe what the slope 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 mobile phone salesman earned \$600 in a particular week during which he sold 26 phones and \$540 in another week during which he sold 20 phones.
Write an equation to represent the weekly earnings of the salesman, y, as a function of the number of phones sold, x.
State the slope of this function.
Describe what the slope of the line represents in context.
Find the y-intercept.
Describe what the y-intercept represents in context.
Find how much the salesman will earn in a week during which he sells 36 phones.
Paul has just purchased a prepaid phone, which he intends to use exclusively for sending text messages, and has purchased some credit along with it to use.
After sending 11 text messages, he has \$34.39 of credit remaining and after sending 19 text messages, he has \$30.31 of credit remaining.
The relationship between the number of text messages sent and the amount of credit remaining is linear. Determine the slope of the linear function.
Write an equation to represent the amount of credit remaining, y, as a function of the number of text messages sent, x.
Describe what the slope of the line represents in context.
State the value of the y-intercept.
State the meaning of the y-intercept in this context.
Find how much credit Paul will have left after sending 36 text messages.
A car travels at an average speed of V = 75\text{ km/h} away from home.
Construct an equation to represent the distance travelled in kilometres, D, away from home after t hours.
Describe what the slope of the line represents in context.
The vehicle has enough petrol to drive a distance of 465\text{ km}. Find the value of t, the time it takes in hours for the car to travel a distance of 465\text{ km} away from home.
By considering the context, state the domain of your equation.
If the car is initially 15\text{ km} away from home, construct another equation to represent the distance travelled in kilometres, D, from home after t hours.
The number of fish in a river is approximately declining at a rate of 200 fish per year.
If there are initially 4000 fish in the river, construct an equation to represent the amount of fish, F, in the river after t years.
Describe what the slope of the line represents in context.
Find the value of t, the time it takes in years for the population of fish in the river to reach zero.
By considering the context, state the domain of your equation.
If there are initially 2400 fish in the river, construct another equation to represent the amount of fish, F, in the river after t years.
Amy is taking her new car out for a drive. She has only driven 50\text{ km} in it previously and is now driving it down the highway at 60\text{ km/h}.
Construct an equation to represent the total distance travelled by the car in kilometres, D, after t hours of driving down the highway.
Describe what the slope of the line represents in context.
The car reaches its destination along the highway. The total milage on the car at this point is 140 \text{ km} km. Find the value of t, the time it takes in hours for the car to reach its destination.
By considering the context, state the domain of your equation.
If the car had previously driven 90 \text{ km} instead, construct another equation to represent the total distance travelled in kilometres, D, after t hours down the highway.
Glen is running a 180 \text{ km} ultramarathon at an average speed of 9 \text{ km/h}.
Construct an equation to represent the total distance travelled by Glen in kilometres, D, after t hours of running in the ultramarathon.
Describe what the slope of the line represents in context.
Find the value of t, the time it takes in hours for Glen to reach the halfway point.
By considering the context, state the domain of your equation.
Construct another equation to represent the total distance travelled in kilometres, D, t hours after reaching the halfway mark.
A carpenter charges a callout fee of \$150 plus \$45 per hour.
Construct an equation to represent the total cost incurred in dollars, C, after t hours worth of work.
Describe what the slope of the line represents in context.
Find the value of t, the time it takes in hours for the carpenter to complete a job that earns \$285.
State the domain of the equation, if the maximum earned from a single job is \$285.
Construct another equation to represent the total cost in dollars, C, after t hours worth of work if the callout fee was instead \$60.