Year 11 General Mathematics Name: _______________ SACE ID: ____________ Mathematical Investigation EXPONENTIAL FUNCTIONS AND GRAPHS Rabbit Populations Exponential functions model growth and decay and have a real place in areas such as science, economics and finance. This investigation aims to look at the various parts of exponential functions to draw conclusions about what happens over time. You will then use exponential functions and graphs to model, interpret and predict the growth and state of rabbit populations in a given scenario. Your information and calculations must be presented in an investigation which includes the following areas: Introduction Mathematical Procedures. You must create your own titles/headings/sentences to explain everything, assume the person marking it does not have the task sheet. Conclusion of results Reference List (with in-text referencing) Maximum number of pages: 8 The performance standards being assessed in this task are: Concepts and Techniques CT1 Knowledge and understanding of concepts and relationships. CT2 Selection and application of mathematical techniques and algorithms to find solutions to problems in a variety of contexts. CT3 Application of mathematical models. CT4 Use of electronic technology to find solutions to mathematical problems Reasoning and Communication RC1 Interpretation of mathematical results. RC2 Drawing conclusions from mathematical results, with an understanding of their reasonableness and limitations. RC3 Use of appropriate mathematical notation, representations, and terminology. RC4 Communication of mathematical ideas and reasoning to develop logical arguments. RC5 Forming and testing of predictions. MATHEMATICAL PROCEDURES Part A – Basic exponential functions 1. A. Create a table of values for the function 𝑦 = 2𝑥 , for 0 ≤ 𝑥 ≤ 6 (values of 𝑥 from 0 to 6). B. Comment on what you notice about the answers. C. Plot the function 2𝑥 by hand on a suitably sized cartesian plane. D. Predict what changes you think will happen to the graph, when the number ‘2’ increases or decreases. E. Graph the following exponential functions using a graphing package eg Desmos, on one cartesian plane, for 0 ≤ 𝑥 ≤ 10. Include a clear screenshot. i. 𝑦 = 2𝑥 iii. 𝑦 = 5𝑥 𝑥 ii. 𝑦 = 1.2 iv. 𝑦 = 10𝑥 F. Describe the shape of each graph and analyse the difference between the graphs. Comment on whether your prediction was correct or not. G. Which equation from above would represent increasing by 20%? 2. A. Create a table of values for the function 𝑦 = 3 × 2𝑥 , for 0 ≤ 𝑥 ≤ 6. How do your answers compare to the table of 𝑦 = 2𝑥 ? B. If in the equation 𝑦 = 3 × 2𝑥 , 𝑦 was the dollar amount in a bank account/investment, and 𝑥 was the number of months that had passed, how much money was initially invested? What happens to the money each month? C. What equation would represent having $10 that triples each month? D. Predict what you think the y-intercept (the starting point, when x=0) will be for the equations below. E. Graph the following exponential functions using a graphing package, on one cartesian plane, for 0 ≤ 𝑥 ≤ 10. i. 𝑦 = 3 × 2𝑥 iii. 𝑦 = 10 × 2𝑥 𝑥 ii. 𝑦 = 0.2 × 2 iv. 𝑦 = 5 × 2𝑥 F. What was the y-intercept of each of the graphs? Was your prediction correct? Discuss whether the shape of each graph has changed or not. 3. A. Create a table of values for the function 𝑦 = 32 × 0.5𝑥 , for 0 ≤ 𝑥 ≤ 6. B. Compare these numbers to your table of values and graph for 𝑦 = 2𝑥 . What do you notice? Extension: Can you explain mathematically what is happening? C. Predict how the graph 𝑦 = 32 × 0.5𝑥 will be different to all the graphs so far. D. Graph the following exponential functions using a graphing package, on one cartesian plane, for 0 ≤ 𝑥 ≤ 10: v. 𝑦 = 32 × 0.5𝑥 vii. 𝑦 = 32 × 0.0001𝑥 𝑥 vi. 𝑦 = 32 × 0.02 viii. 𝑦 = 32 × 0.9𝑥 E. What impact does having the base number as a decimal, ie. 0 < 𝑏 < 1 have on the graph? Was your prediction correct? F. What equation would represent $70 decreasing by 5% each month? Extension There are 2 options for extension. Both options require using a similar process to the questions above, eg making a prediction, graphing a few on desmos, and commenting on the results. Option 1 is to investigate the effect of having a negative out the front of the equation eg = −2𝑥 . Option 2 is to investigate adding and subtracting numbers to the equation eg 𝑦 = 2𝑥 + 5 Part B – Real life applications – Rabbit Populations Scenario: Rabbits are a real pest in Australia. They breed very quickly and have no natural predators in most regions of Australia. The introduction of rabbits began in 1859, when a farmer who had migrated from England, felt homesick and imported 24 wild English rabbits and set them free on his land. After only 6 years (or 72 months), his rabbit population had multiplied to millions (http://www.petefalzone.com/handouts/exp-growth-rabbits-australia.pdf). It is said that the population growth of rabbits at the time increased at a rate of 19% each month, but this figure is not an entirely accurate picture, as their population compounds so often. Note any answers that come up in scientific notation should be written in this format and to 3 significant figures: 3.45 × 109 . If the number is not too big, write it also as a basic numeral to help show the size of the number. 1. Given the rabbit population started at 24, if it doubled every month, write an equation that expresses this situation. 2. What would the rabbit population be after: a. 6 months b. 1 year c. 2 years? 3. It has been discovered that the rabbit population (24) actually only increases by 0.615% each day. Write the new equation that expresses this situation. Take care when converting this percentage to a decimal. 4. What would the rabbit population be: a. After 30 days? b. After 1 year? c. After 10 years? d. After 30 years? 5. After 30 years, the population levelled off and stopped increasing. It didn’t drop; rabbits were merely reproducing at the same rate as they were dying. There were approximately 750,000,000 rabbits at this point. a. Think of some reasons why this number is different to your answer above. What assumptions have we made in those calculations? What limitations are there to using an equation to predicting the number of rabbits? b. Research or come up with some reasons why the population growth slowed down. Extension One of these reasons was the Myxomatosis virus, introduced in Australia in the early 1950s, which is lethal to rabbits. This was done to reduce numbers easily and with little intervention. a. Assuming there are 750,000,000 rabbits and it decreased the rabbit population by 6% each month (despite breeding), use an exponential equation and graph to show the population decline. b. Explain what is happening over time by analysing your results. c. Will the population ever reach zero? Explain. d. What assumptions and limitations are there to your calculations? e. What are some issues with using a virus as a method of rabbit control? CONCLUSION Summarise your results and write a conclusion that covers both Part A (patterns in exponential functions) and Part B (what happens with the rabbits). Stage 1 General Mathematics – Performance Standards - Concepts and Techniques Reasoning and Communication A Comprehensive knowledge and understanding of concepts and relationships. Comprehensive interpretation of mathematical results in the context of the problem. Highly effective selection and application of mathematical techniques and algorithms to find efficient and accurate solutions to routine and complex problems in a variety of contexts. Drawing logical conclusions from mathematical results, with a comprehensive understanding of their reasonableness and limitations. Successful development and application of mathematical models to find concise and accurate solutions. Appropriate and effective use of electronic technology to find accurate solutions to routine and complex problems. B Mostly effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine and some complex problems in a variety of contexts. Drawing mostly logical conclusions from mathematical results, with some depth of understanding of their reasonableness and limitations. Mostly accurate use of appropriate mathematical notation, representations, and terminology. Mostly effective communication of mathematical ideas and reasoning to develop mostly logical arguments. Formation and testing of mostly appropriate predictions, using some mathematical evidence. Generally competent knowledge and understanding of concepts and relationships. Generally appropriate interpretation of mathematical results in the context of the problem. Generally effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine problems in different contexts. Drawing some logical conclusions from mathematical results, with some understanding of their reasonableness and limitations. Application of mathematical models to find generally accurate solutions. Generally appropriate and effective use of electronic technology to find mostly accurate solutions to routine problems. Basic knowledge and some understanding of concepts and relationships. Some selection and application of mathematical techniques and algorithms to find some accurate solutions to routine problems in context. Some application of mathematical models to find some accurate or partially accurate solutions. Some appropriate use of electronic technology to find some accurate solutions to routine problems. E Formation and testing of appropriate predictions, using sound mathematical evidence. Mostly appropriate interpretation of mathematical results in the context of the problem. Mostly appropriate and effective use of electronic technology to find mostly accurate solutions to routine and some complex problems. D Highly effective communication of mathematical ideas and reasoning to develop logical and concise arguments. Some depth of knowledge and understanding of concepts and relationships. Attempted development and successful application of mathematical models to find mostly accurate solutions. C Proficient and accurate use of appropriate mathematical notation, representations, and terminology. Limited knowledge or understanding of concepts and relationships. Attempted selection and limited application of mathematical techniques or algorithms, with limited accuracy in solving routine problems. Attempted application of mathematical models, with limited accuracy. Attempted use of electronic technology, with limited accuracy in solving routine problems. Generally appropriate use of mathematical notation, representations, and terminology, with reasonable accuracy. Generally effective communication of mathematical ideas and reasoning to develop some logical arguments. Formation of an appropriate prediction and some attempt to test it using mathematical evidence. Some interpretation of mathematical results. Drawing some conclusions from mathematical results, with some awareness of their reasonableness. Some appropriate use of mathematical notation, representations, and terminology, with some accuracy. Some communication of mathematical ideas, with attempted reasoning and/or arguments. Attempted formation of a prediction with limited attempt to test it using mathematical evidence. Limited interpretation of mathematical results. Limited understanding of the meaning of mathematical results, their reasonableness or limitations. Limited use of appropriate mathematical notation, representations, or terminology, with limited accuracy. Attempted communication of mathematical ideas, with limited reasoning. Limited attempt to form or test a prediction.
