Exponential Growth and Decay

Exponential Growth and Decay

What does exponential growth and decay look like on a graph?

• Not all functions increase by the same rate of change value/slope

• Only linear functions have a constant rate of change

• For linear functions, the graph forms a straight line, either rising or falling from left to right

• Exponential growth and decay are exponential functions, forming a curved line. If it rises from left to right it is growth. If it falls from left o right it is decay.

Exponential Function Review

• Equation is f(x) = a $b^x$  , where b $>$ 1. 

• Why does b have to be > 1?  1 to any power is always 1.

• "a" is the initial value when x = 0. It can be 1, but it cannot be 0.

• The rate of change is measured in the quotients of the y-values, not the differences of the y-values. The quotient will be the value of "b".
  

Exponential Growth

• f(x) = (a) $b^x$  , where b>1, a  $\ne$  0.

• growth rate = rate at which the function is growing

• rate can be given in percent, but is always converted to its decimal form to be used algebraically.

• growth factor = 1 + (growth rate)

• b = growth factor

Examples of Exponential Growth

Microorganisms in cultures

Pandemics, like corona virus

Food Spoilage

Compound Interest

Example of Exponential Growth Problem

• Initial value = 77

• Growth factor is given = 2 (doubles)

• Time = x = 17 weeks/2 (every 2 weeks it doubles = 17/2

• population in 17 weeks = initial value (growth factor) raised to the time divided by time it takes to double

• Final equation: f(x) = 77(2)^17/2 = 27,877 zombies

Jordan plants a tree that is 5 feet tall. How tall will the tree be in 15 years if it grows at a rate of 10% each year?

• Initial height = 5 feet = a

• Growth rate = 10% = .10

• Time = 15 years; Grows 10% each year, so time = 15/1 = 15

• Growth factor = 1 + rate = 1 + .10 = 1.1

• Height after 15 years = f(x)

• f(x) = initial value (growth factor)^time

• f(x) = 5 (1.1)¹⁵ = 20.886 feet

What would happen if we kept growing at the same rate until we die?????

• Suppose we start out life at 21 inches and grow at a rate of 10% per year

• growth rate = 10% = .1; growth factor = 1 + .1 = 1.1

• let's say life expectancy is 85 years

• Height in 85 years = (21) (1.1)⁸⁵ = 69, 278.35

• That's inches! Divide by 12 to find the number of feet.

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• Okay, that is still 5773.2 feet. We would be very tall people if we didn't stop growing!!!!!

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How to calculate exponential growth

• 1. Determine the initial value. This is the value when x =0, or the amount you start out with.

• 2. If the growth rate is given as a percent, convert to decimal form.

• 3. Determine the growth factor. Growth factor is (1 + growth rate)

How to calculate Exponential Growth

• 4. Determine time. Time = total time / repeats per cycle

• Doubling every other week for 29 weeks would be 29/2

• 5. Substitute the values into the formula: f(x) = a (b)^x, where a = initial value, b = growth factor, and x = time

• In the example, a = 5000, rate = 5.5% = 0.055, growth factor = b = 1+0.055, and time = t (unknown)

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Things to remember:

• The growth factor will be greater than 1 for exponential growth

• To find the growth rate if you are given the factor, subtract 1 from the growth factor. If the factor is 1.76, the growth rate is 1.76 - 1, or .76, or 76%

• Always add 1 to the growth rate to get the growth factor, unless it is given to you as a growth factor (i.e., doubling, tripling, increases 2 times)

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Multiple Choice

There were 417 cell phones sold at an electronics store in January. Since then, cell phone sales at this store have increased at a rate of 3.75% per month.

At this rate of growth, which function can be used to determine the monthly cell phone sales x months after January?

Hint: Formula is y = a (b)x. a = initial value, b = growth factor, x = time. Growth factor is 1 + growth rate (change rate to decimal)

• $f(x)=417(1-0.0375)^x$

• $f(x)=417(1-3.75)^x$

• $f(x)=417(1+0.0375)^x$

• $f(x)=417(1+3.75)^x$

Multiple Choice

What type of function is y = 7(5/4)^x?

Hint: Is the growth factor > 1, or <1?

• Exponential Growth

• Exponential Decay

• Linear

• None of the above

Multiple Choice

Jame's 70 in. giant peach doubles in size every week. Write an expression that would represent how big the peach is after 5 weeks.

Hint: Hint: Formula is y = a (b)^x. a = initial value, b = growth factor, x = time. Growth factor is 1 + growth rate (change rate to decimal)

• 70(2)³⁵

• 70(2)⁵

• 2(70)⁵

• 5(70)²

Multiple Choice

The number of mosquitoes at the beginning of the summer was 4,000. The population of mosquitoes is expected to grow at a rate of 25% a month. How many mosquitoes will there be after 4 months?

Hint: Formula is y = a (b)^x. a = initial value, b = growth factor, x = time. Growth factor is 1 + growth rate (change rate to decimal)

• 9766

• 9006

• 9765

• 5433

Multiple Choice

Is this exponential growth or decay?

• Growth

• Decay

• Linear

Multiple Choice

What is a, the initial amount, for the function: f(x) = 300(1.16)^x?

• 300
• 1.16
• .16
• x

Multiple Choice

Write an equation that models the following situation:

Samantha's hair was known to grow very rapidly. It began at a length of 6 in and grew at a rate of 14% a week.

Hint: initial value is what her hair started at; change the rate to a decimal; for growth factor add 1 + growth rate); x = time. Formula is f(x) = y = a (b)^x, where a = initial value, b = growth factor, exponent is the variable.

• y=6(0.14)^x

• y=6(1+14)^x

• y=6(1.14)^x

• y=6(0.86)^x

Multiple Choice

What is r, the growth rate, for the function: f(x) = 300(1.16)^x?

Hint: Since the b value (growth or decay factor) is greater than 1, we know it is exponential growth. To find the growth factor, you add 1 to the growth rate, factor = 1 + r. Work backwards to solve for r. Growth factor - 1 = rate. Convert to percent.

• 30%

• 116%

• 16%

• x%

Exponential Decay

• Formula is the same as exponential growth (y = a (b)^x ) EXCEPT

• Instead of adding the rate of growth/decay to 1, you SUBTRACT it from 1.

• Decay factors will always be less than 1; growth factors will always be greater than 1

• If the decay rate is 15%, the decay factor is 1 - .15, or .85

How to Set Exponential Decay Problems

• 1. Determine the initial value. This is what you start with, and the value when x = 0. In the example the initial value is $23,000.

• 2. Determine the decay rate. If the decay rate is given as a percent, change to a decimal. In the example the decay rate is 15% or .15

• 3. Determine the decay factor. The decay factor is (1 - decay rate), In the example the decay factor is 1 - .15 or .85

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• 4. Determine the time. This would be the number of days, months, years, etc. If your item decays more than 1 unit/decay, divide the time by that number. In the example the time is the variable "t"

• ie. If it decays 10% every 2 years, then you would express time as t/2, or x/2

• 5. Substitute your values into the formula: f(x) = y = (a) (b)^x, where a = initial value, b = decay factor, and x = time: f(x) = 23000 (.85)^t

Exponential Growth: Factor > 1; Exponential Decay: Factor <1 (but greater than 0)

All exponential decay functions will have this downward curve

Examples of Exponential Decay from Real Life

Car Depreciation

• The minute you drive a car off the sales lot, it begins to depreciate (lose value)

• The "Blue Book" value of a car tells you your cars value based on depreciation and the condition the car is in (excellent, good, fair, poor)

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Radioactive Decay

Decrease in Population in a Community/Country

• Populations within a given area can decrease

• This happens in rural areas when the young people leave after graduation and do not return to the area

• This can also happen in a city when an industry shuts down

• There are more people moving out than are staying and repopulating

Atmospheric Pressure Decreases as the Altitude Increases

• Atmospheric pressure (the pressure of air around you) decreases as you go higher.

• It decreases about 12% for every 1000 m: an exponential decay.

• The pressure at sea level is about 1013 hPa (depending on weather). (from MathBits)

• y = 1013 (1-.12)^x = 1013(.88)^x

Multiple Choice

A population of 1500 deer decreases by 1.5% per year. At the end of 10 years, there will be approximately 1290 deer in the population.

Which function can be used to determine the number of deer, y, in this population at the end of t years?

Hint: The formula for decay is y = a $b^x$  where a = initial value, b = 1 - decay factor, and x = time

• $y=1500(1-0.015)^t$

• $y=1500(0.015)^t$

• $y=1500(1+0.015)^t$

• $y=1500(1.5)^t$

Multiple Choice

Suppose a culture of bacteria begins with 5000 cells and dies by 30% each year. Write an equation that represents this situation.

Hint: The formula for decay is y = a b^x

where a = initial value, b = 1 - decay factor, and x = time

• y=5000(0.7)^x

• y=30(5000)^x

• y=5000(1.3)^x

• y=5000x^x

Multiple Choice

A 78 gram sample of Uranium loses half of its mass each year. What is the exponential equation? 
Hint: The formula for decay is  y = a  $b^x$    

where a = initial value, b = 1 - decay factor, and x = time

• y=78(0.5)^x

• y=78(1.5)^x

• y=78(2)^x

Multiple Choice

Classify the model as Exponential GROWTH or DECAY.

A=1200(.85)⁶

Hint: Is the b value (inside the parentheses) greater than 1, or less than 1?

• Growth

• Decay

Multiple Choice

What is the decay rate in the following model?

A=1200(.85)⁶

Hint: If the b factor (inside the parentheses) is less than 1, it is decay, and the rate was subtracted from 1 to get the decay factor. In order to find the rate, work backwards. Factor = 1 - rate. Then 1-factor = rate. Change to a percent.

• 85%

• 15%

• 100%

• 1.5%

Extra Resources

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