Sketch the following piecewise functions:
Consider the following piecewise function:
y = \begin{cases} -4x-6, & x\lt 0 \\ x+3, & x \gt 0 \end{cases}If the graph of the piecewise function is connected, find the value of a, by solving the equations simultaneously.
What is the y-value of the piecewise function at x = a?
Write the piecewise function definition of the following graphs:
The following line graph shows the distance Buzz was from his house as he travelled to visit his grandparents:
Given that Buzz left home at 10 am, find:
How far Buzz was from his house at 10:30 am.
The time Buzz got back home.
The furthest distance Buzz was from his house.
A kitchen sink starts with 20 L of water. It empties at a rate of 4 L per minute and then the drain gets blocked. No water empties for 2 more minutes while it gets unblocked. Then the remaining water is drained out in 1 minute.
Construct a piecewise graph that represents this scenario.
A tank is being filled at a constant rate of 4 L per second for 4 minutes. The hose is then turned off for 2 minutes. Finally it is topped up again with water at a rate of 3 L per second for 1 minute.
Construct a piecewise graph that represents this scenario.
Danielle's distance from home throughout the day is recorded on the line graph below:
How far was Danielle from her house at 2:00 pm?
What time did Danielle get back home?
What was the furthest distance Danielle was from her house?
The line graph shows the amount of petrol in a car’s tank during a long drive:
Given that the drive started at 8 am:
How much petrol was initially in the tank?
What happened at 9 am and 1 pm?
How much petrol was used between 1 pm and 5 pm?
To the nearest hour, when did the petrol in the tank first fall below 18 litres?
A children's pool is being filled with water. The volume of water V, in litres, after t minutes is given by the piecewise graph below. The pool has a maximum capacity of 1500L.
Estimate the volume of water in litres after 45 minutes have elapsed.
Determine the equation that describes the volume of water V in litres after t minutes for the first 45 minutes.
After 45 minutes, the rate at which the volume of water enters the pool is increased. The piecewise relationship after t = 45 is given by V = 26 t - 970.
Determine the value of t (in minutes) at the time when the pool is filled up.
A marathon runner initially runs at an average pace of 15 km/h. After 40 minutes of running at this speed, the runner slows down to a more comfortable pace of 12 km/h for the remaining 160 minutes.
Let D be the distance covered by the marathon runner in kilometres and t be the time elapsed in minutes.
Define the piecewise function for D in terms of t.
Find the distance covered in kilometres after:
20 minutes
40 minutes
60 minutes
200 minutes
Sketch the graph of the piecewise function.
Harry goes out for a run. He accelerates from rest up to a desired speed and maintains that speed for some time. Feeling exhausted, his velocity drops until he's back at rest.
The speed S in km/h after t seconds is given by the following piecewise relationship:
S = \begin{cases} 0.8t, & 0 \leq t \lt 15 \\ 12, & 15 \leq t \lt 240 \\ 252 - t, & 240 \leq t \leq 252 \end{cases}Find the speed in km/h after:
10 seconds
50 seconds
150 seconds
250 seconds
Sketch the graph of the piecewise function.
The amount of energy stored E in kWh in a set of solar batteries t hours for a 36 hour period after 6:00 am is given by the piecewise graph below:
Part of the piecewise graph is given by E = \dfrac{7}{6} t, \, 0 \leq t \leq a. Determine the value of a.
After 12 hours have elapsed, the amount of energy stored is 14 kWh. Determine the equation for the part of the piecewise graph over the region
12 < t \leq 24.
The equation that describes the last piece of the piecewise graph is
E = t - 10. State the time interval that the last piece is defined over.
Consider the following step graph:
State the equation of the line for x > 3.
State the equation of the line for x \leq 3.
Consider the following step graph:
State the equation of the line for
x > - 5.
State the equation of the line for
x \leq - 5.
Consider the following step graph:
Define the function for x > 4.
Define the function for x \leq 4.
The graph shows the cost, in dollars, of sending parcels of various weight overseas:
Find the cost of sending a letter weighing 100 grams.
Find the cost of sending a letter weighing 300 grams.
What is the heaviest letter that can be sent for \$2?
The graph shows the parking costs over different lengths of time:
How much would 5 hours of parking cost?
What is the longest you can park your car for \$6?
The carpark offers a weekly pass for \$44. If Aaron parks his car for 6 hours each day, five days a week, how much would he save each week with the weekly pass?
The graph shows the amount (in dollars) an internet cafe charges its customers:
How much does Parvaneh have to pay if she uses their internet service for 2 hours and 40 minutes?
How much does Parvaneh have to pay if she uses their internet service for 4 hours?
What is the maximum number of hours that Parvaneh can use the internet service for \$15?
The graph shows the starting times of a shot put event for participants of different age groups:
What time will a participant who is aged 13 start their race?
Do any of the races start at half past the hour?
The graph shows the total cost of parking (in dollars) as a function of the number of hours parked:
How many free hours does this parking garage offer?
Peyvand goes to a movie and parks for 2.5 hours. How much will it cost her?
Tirdad works at the cinemas and parks for his entire 4.5 hour shift. How much will it cost him in parking?
The graph shows the amount (in dollars) a lawyer charges for consultations:
How much does the lawyer charge for a 5.5 hour consultation?
How much does the lawyer charge for a 4 hour consultation?
What is the shortest possible consultation that the lawyer will charge \$800 for?
The graph shows the cost per t-shirt (in dollars) when purchasing in bulk:
What is the cost per t-shirt when 45 t-shirts are bought in bulk?
What is the cost per t-shirt when 55 t-shirts are bought in bulk?
What is the most number of t-shirts that can be purchased for \$7.00 each?
The graph shows the cost (in dollars) of a mobile phone call as a function of the length of the call:
How much does a call that lasts 4 minutes and 5 seconds cost?
How much does a 3-minute call cost?
What is the longest possible call that could be made for \$1.50?
What is the cost of each additional minute?
What is the connection cost?
The step graph shows the cost, in dollars, for postage of packages based on their weight:
Find how much it will cost to post a package that weighs:
1.7 kg
2.5 kg
0.5 kg
\$3.2 kg
If Neil has \$11, what is the weight of the heaviest package he can send?
Find the heaviest a package can be if it is sent for:
The cost of a train ticket, based on distance travelled (in km), is indicated by the step graph:
If Valentina needed to travel a distance of 11 km, how much would her train ticket cost?
Luke has a doctor's appointment 19 km away. How much money will he need for a round-trip ticket (there and back again)?
Sally needs to travel 4 km to get to the grocery store. How much will a one way ticket cost?
With \$3, if Dave buys a round-trip ticket (there and back again), what is the furthest he can travel?
Maria commutes approximately 40 km each direction every day to work. Calculate the total cost for one week's worth (5 days) of round-trip tickets.
The step graph shows the charges for a home phone plan:
How much will a 3-minute phone call cost?
How much will 4 phone calls, each longer than 1 minute but less than 2 minutes, cost?
A phone line user made the following calls during one month:
12 calls that were just under 2 minutes long each
13 calls just under 4 minutes long each
5 calls of less than 1 minute each
Find the total charges for the month to the nearest dollar.
Hiring helium tanks for balloons at a party costs \$20 up to and including the first hour, and \$5 per extra hour or any part thereof. Sketch the step graph that represents the total cost for any number of hours.
A telephone call is charged at \$0.70 for a call length of less than 1 minute, and \$0.20 extra per minute after that. Graph the cost of calls up to 5 minutes in length, where the vertical axis is in dollars.
The cost of a train fare is calculated at \$1.30 for the first zone and then 20c per additional zone after that. Graph the fares for trips up to five zones, with the vertical axis in dollars.
At an indoor ski facility, the temperature is set to - 5 \degreeC at 2 pm. At 3 pm, the temperature is immediately brought down to - 12 \degreeC and left for 3 hours before immediately taking it down again to - 18 \degreeC, where it stays for the rest of the day’s operation. The facility operates until 10 pm.
Write a stepwise function that models the indoor temperature, y, at a certain time of the day, x hours after midday.
Create a graph of the step function relating time of day and temperature inside the ski facility.
Lakota entered the ski facility at 3:30 pm. What was the temperature inside the facility at this time?
Xavier wants to wait till the indoor temperature is - 7 \degreeC or lower. When is the earliest he can enter the facility?
A plumber charges a call-out fee of \$20 to come out to the site and on top of that, charges \$10 per 15 minute block on site or part thereof. Create a step graph that matches this scenario, with the vertical axis in dollars.
Create a step graph for the charge for sending parcels, where the vertical axis is in dollars.
| Express Post satchels (mm) | Maximum weight (g) | Charge per item |
|---|---|---|
| \text{Small }(220 \times 355) | 500 | \$2.50 |
| \text{Medium }(310 \times 405) | 3000 | \$4.25 |
| \text{Large }(435 \times 510) | 5000 | \$7.75 |
The government is looking at alternative taxation schemes and one proposal is to pay a fixed amount of tax depending on which income bracket you fall into.
The function below models the tax payable, y, based on an income of x:
y = \begin{cases} \$1200, & \$0 \lt x \leq \$15\,000 \\ \$4900, & \$15\,000 \lt x \leq \$35\,000 \\ \$12\,350, & \$35\,000 \lt x \leq \$65\,000 \\ \$43\,320, & \$65\,000 \lt x \leq \$114\,000 \\ \$54\,720, & x \gt \$114\,000 \\ \end{cases}Under this proposal, how much tax will you pay on an income of \$14\,000?
What is the difference, in whole dollars, in income between the highest income earner and the lowest income earner who are both paying \$43\,320 in tax?
Estelle is earning an income of \$63\,800. She is set to get a pay rise at the end of the year. What is the maximum pay rise she can get before falling into the next highest tax bracket?