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Exponential Functions and Graphs

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Exponential Functions and Graphs

A community recycling drive is tracking how its online sign-ups and uncollected waste change over time. Use exponential functions, tables and graphs to analyse the data. Give exact values where possible and round decimals to 2 decimal places.

Recognise and represent

1.The number of people signed up is modelled by S(t) = 80(1.25)^t, where t is the number of weeks since launch. Which statement best describes the model?

The number increases by 25 people each week.
The number is multiplied by 1.25 each week, so it grows by 25% per week.
The number decreases by 25% each week.
The number starts at 1.25 people.

2.For S(t) = 80(1.25)^t, state the initial number of sign-ups, the weekly growth rate as a percentage, and the number predicted after 4 weeks.

3.A recycling team records 120 kg of material remaining to be sorted. Each day, the remaining amount is 70% of the previous day's amount. Write a function R(d) for the amount after d days, and identify the decay rate.

Tables and graph features

4.Complete the value table for f(x) = 2^x using the given x-values. Then draw Cartesian axes with x-values from -2 to 2 and y-values from 0 to 4, plot the points, and sketch a smooth curve.

f(x) = 2^x

-2

-1

5.For g(x) = (1/2)^x, calculate g(x) for x = -2, -1, 0, 1, 2. State whether it shows growth or decay, and give its horizontal asymptote.

6.Which features are true for y = a^x when a > 1? Select all that apply.

The y-intercept is (0, 1).
The graph has horizontal asymptote y = 0.
The graph passes through (1, 0).
The function is increasing for all real x.
The range is y > 0.

7.Compare y = 3^x with y = 2^x for x > 0. Which graph rises more steeply, and what do they have in common?

Transformations and applications

8.The model h(x) = 2^x + 3 is made from y = 2^x. Describe the transformation, the horizontal asymptote, and the y-intercept.

9.Sign-ups are modelled by S(t) = 80(1.25)^t. How many sign-ups does the model predict after 6 weeks? Round to the nearest whole person.

10.The sorting team has 120 kg remaining and models it by R(d) = 120(0.7)^d. After how many whole days will less than 30 kg remain? Show how you decide.

11.Freya says that 80(1.25)^t grows by 25 sign-ups each week because 1.25 is the growth amount. Explain the error and state the increase from week 0 to week 1.

12.Write one realistic example from the recycling drive that could be modelled by exponential decay. Define the starting amount, the time interval and the multiplier in your model.