Mathematics
Grade 9
15 min
Describe linear and exponential growth and decay
Describe linear and exponential growth and decay
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify if a pattern in a table represents linear or exponential change.
Calculate the common difference for linear functions and the common ratio for exponential functions.
Write equations in the form y = mx + b and y = a(b)^x to model real-world scenarios.
Distinguish between growth (increasing) and decay (decreasing) for both linear and exponential models.
Interpret the meaning of the parameters (initial value, rate of change, growth/decay factor) in the context of a problem.
Compare linear and exponential models to determine which grows or decays faster over a given interval.
Would you rather have a magic pot that gives you $100 every day, or one that starts with $1 and doubles its contents each day? 🤔 The answer might surprise you!
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Key Concepts & Vocabulary
TermDefinitionExample
Linear Growth/DecayA pattern where a quantity changes by adding or subtracting the same constant amount in each time interval.Your savings account has $50, and you deposit $10 each week. The amount grows linearly: $50, $60, $70, $80...
Exponential Growth/DecayA pattern where a quantity changes by being multiplied by the same constant factor in each time interval.A social media post has 100 likes. The number of likes doubles every hour. The amount grows exponentially: 100, 200, 400, 800...
Common Difference (d or m)The constant amount that is added (for growth) or subtracted (for decay) in a linear pattern.In the sequence 12, 9, 6, 3..., the common difference is -3.
Common Ratio (r or b)The constant factor that you multiply by in an exponential pattern. If r > 1, i...
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Core Formulas
Linear Function Form
y = mx + b
Use this to model linear growth or decay. 'm' is the common difference (rate of change), and 'b' is the initial value (the y-intercept).
Exponential Function Form
y = a(b)^x
Use this to model exponential growth or decay. 'a' is the initial value, and 'b' is the common ratio (growth or decay factor).
Calculating the Decay Factor
b = 1 - r
To find the decay factor 'b' for an exponential function, convert the percentage decrease 'r' to a decimal and subtract it from 1.
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Challenging
Car A is bought for $30,000 and loses 20% of its value each year. Car B is bought for $25,000 and loses 15% of its value each year. After 3 years, which car has a higher value?
A.Car A
B.Car B
C.Their values are equal
D.Cannot be determined
Challenging
The population of a town is growing exponentially. In 2015 (year x=0), the population was 12,000. In 2020 (year x=5), the population was 18,000. Which equation best models the town's population?
A.y = 12000(1.084)^x
B.y = 12000(1.5)^x
C.y = 18000(1.084)^x
D.y = 12000(1.1)^x
Easy
Which of the following patterns describes linear growth?
A.quantity is multiplied by 3 in each time interval.
B.quantity increases by adding the same constant amount in each time interval.
C.quantity decreases by being multiplied by a factor of 0.5 in each time interval.
D.quantity changes by adding a different amount in each time interval.
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