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Graphs of practical situations

Construct and interpret quadratic (y=ax2y = ax^2), exponential (y=abxy = ab^x) and reciprocal (y=k/xy = k/x) graphs in practical contexts; read maximum/minimum values, intercepts and rates of change from real-world graphs.

Worked examples

Quadratic model — finding a maximum

Straightforward

Problem

A ball is thrown upward. Its height hh (metres) at time tt (seconds) is h=−5t2+20th = -5t^2 + 20t. Find (a) the height at t=1t = 1 s and (b) the maximum height.

Exponential model — growth and decay

Moderate

Problem

A car purchased for $30\,000 depreciates at 15% per year. Its value after tt years is V=30 000×(0.85)tV = 30\,000 \times (0.85)^t. (a) Find the value after 4 years. (b) After how many years is the car worth less than $10\,000?

Reciprocal model

Challenging

Problem

The time tt (hours) to complete a job varies inversely with the number of workers nn: t=24nt = \dfrac{24}{n}. (a) How long does the job take with 6 workers? (b) How many workers are needed to complete it in 3 hours?

Practise

Q1·Straightforward
A ball is thrown upward and its height is modelled by h=−t2+4th = -t^2 + 4t (metres), where tt is the time in seconds. Find the height of the ball at t=2t = 2 seconds.
Q2·Straightforward
The temperature of a room (°C) after switching off the heating is modelled by T=20×(0.9)tT = 20 \times (0.9)^t, where tt is time in hours. Find the temperature after 2 hours.
Q3·Straightforward
A reciprocal relationship is given by y=12xy = \dfrac{12}{x}. Find the value of yy when x=3x = 3.
Q4·Straightforward
The quadratic curve y=x2−4y = x^2 - 4 crosses the yy-axis at one point. Find the yy-intercept.
Q5·Moderate
The height (m) of a ball thrown upward is modelled by h=−2t2+8th = -2t^2 + 8t, where tt is time in seconds. Find the maximum height reached by the ball.
Q6·Moderate
The cost CC (dollars) of producing xx items is modelled by C=2x2+100C = 2x^2 + 100. Find the cost of producing 10 items.
Q7·Moderate
A bacteria population grows according to N=500×2tN = 500 \times 2^t, where tt is time in hours. How many bacteria are there after 3 hours?
Q8·Moderate
The point (3,8)(3, 8) lies on the reciprocal curve y=kxy = \dfrac{k}{x}. Find the value of kk.
Q9·Moderate
A cooling coffee has temperature T=80×(0.8)tT = 80 \times (0.8)^t degrees Celsius, where tt is time in minutes. Find the temperature after 5 minutes. Give your answer correct to 1 decimal place.
Q10·Moderate
For the reciprocal curve y=6xy = \dfrac{6}{x}, find the value of xx when y=2y = 2.
Q11·Challenging
The profit PP (dollars) from selling xx units is modelled by P=−3x2+60x−200P = -3x^2 + 60x - 200. Find the maximum profit.
Q12·Challenging
A town's population is modelled by P=1000×(1.05)tP = 1000 \times (1.05)^t, where tt is the number of years after 2020. Find the population in 2030 (when t=10t = 10). Give your answer to the nearest whole number.
Q13·Challenging
A radioactive substance decays according to A=1000×(0.5)tA = 1000 \times (0.5)^t (milligrams), where tt is time in hours. After how many hours will 125 mg remain?