Types of relationships

Construct and interpret graphs of practical non-linear relationships; solve simultaneous linear equations graphically and algebraically; model and interpret break-even points and real-world applications.

Algebra · MS-A4

Sub-topics

Graphs of practical situations

Construct and interpret quadratic (

y = ax^2

), exponential (

y = ab^x

) and reciprocal (

y = k/x

) graphs in practical contexts; read maximum/minimum values, intercepts and rates of change from real-world graphs.

Simultaneous linear equations

Solve pairs of linear equations graphically and algebraically; identify the intersection point as a break-even point; model and interpret practical problems involving cost versus revenue and payment plans.

Open Math

Types of relationships

Algebra · MS-A4

Name:

Date:

Graphs of practical situations

Q1Straightforward

A ball is thrown upward and its height is modelled by

h = -t^2 + 4t

(metres), where

t

is the time in seconds. Find the height of the ball at

t = 2

seconds.

Q2Straightforward

The temperature of a room (°C) after switching off the heating is modelled by

T = 20 \times (0.9)^t

, where

t

is time in hours. Find the temperature after 2 hours.

Q3Straightforward

A reciprocal relationship is given by

y = \dfrac{12}{x}

. Find the value of

y

when

x = 3

Q4Straightforward

The quadratic curve

y = x^2 - 4

crosses the

y

-axis at one point. Find the

y

-intercept.

Q5Moderate

The height (m) of a ball thrown upward is modelled by

h = -2t^2 + 8t

, where

t

is time in seconds. Find the maximum height reached by the ball.

Q6Moderate

The cost

C

(dollars) of producing

x

items is modelled by

C = 2x^2 + 100

. Find the cost of producing 10 items.

Q7Moderate

A bacteria population grows according to

N = 500 \times 2^t

, where

t

is time in hours. How many bacteria are there after 3 hours?

Q8Moderate

The point

(3, 8)

lies on the reciprocal curve

y = \dfrac{k}{x}

. Find the value of

k

Q9Moderate

A cooling coffee has temperature

T = 80 \times (0.8)^t

degrees Celsius, where

t

is time in minutes. Find the temperature after 5 minutes. Give your answer correct to 1 decimal place.

Q10Moderate

For the reciprocal curve

y = \dfrac{6}{x}

, find the value of

x

when

y = 2

Q11Challenging

The profit

P

(dollars) from selling

x

units is modelled by

P = -3x^2 + 60x - 200

. Find the maximum profit.

Q12Challenging

A town's population is modelled by

P = 1000 \times (1.05)^t

, where

t

is the number of years after 2020. Find the population in 2030 (when

t = 10

). Give your answer to the nearest whole number.

Q13Challenging

A radioactive substance decays according to

A = 1000 \times (0.5)^t

(milligrams), where

t

is time in hours. After how many hours will 125 mg remain?

Simultaneous linear equations

Q14Straightforward

Solve the simultaneous equations

x + y = 12

and

x - y = 4

. Find the value of

x

Q15Straightforward

Solve the simultaneous equations

2x + y = 9

and

x + y = 6

. Find the value of

x

Q16Straightforward

A company's revenue is

R = 6q

dollars and its total cost is

C = 2q + 20

dollars, where

q

is the number of units sold. Find the break-even quantity (the value of

q

where

R = C

Q17Moderate

Phone Plan A charges $0.40 per minute plus $15 per month. Phone Plan B charges $0.80 per minute plus $5 per month. After how many minutes of calls per month do the two plans cost the same amount?

Q18Moderate

Solve the simultaneous equations

3x - y = 7

and

x + y = 5

. Find the value of

x

Q19Moderate

Two numbers have a sum of 50 and a difference of 8. Find the larger of the two numbers.

Q20Moderate

A business has revenue

R = 10n

dollars and total cost

C = 6n + 48

dollars, where

n

is the number of items produced. How many items must be produced to break even?

Q21Moderate

Two lines intersect at a point. The lines have equations

y = 4x - 3

and

y = x + 6

. Find the

y

-value at the point of intersection.

Q22Challenging

A school canteen sells drinks at $3 each and snacks at $5 each. On one day they sell 60 items in total and take $236. How many drinks were sold?

Q23Challenging

A plumber charges a $200 call-out fee plus $15 per hour. An electrician charges no call-out fee but $40 per hour. After how many hours of work do they charge the same total amount?

Q24Challenging

Solve the simultaneous equations

2x + 3y = 16

and

x - y = 3

. Find the value of

x + y

Q25Challenging

A plumber charges $80 call-out plus $60 per hour. An electrician charges $50 call-out plus $75 per hour. After how many hours does the electrician become more expensive than the plumber? Give the number of hours at which they cost the same.

Worked solutions and answers at openmath.au/year-12/standard-2/types-of-relationships