1 8.1 exponential growth 2 a review of linear growth in algebra 1, almost every real life situation...

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8.1 Exponential Growth

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A Review of Linear Growth

In Algebra 1, almost every real life situation that you encountered was linear. This means that to make mathematical predictions about the situation, you used a linear equation such as y = mx + b.

Linear Growth:

bxay

Initial Amount (Starting Point)

Amount of change (Slope)

Linear Growth assumes that the amount of change during each period is the same. This is rarely true in real-life.

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A Review of Linear Growth

Example (from Algebra 1):

An airplane is flying at an altitude of 5000 feet. As it prepares to land, it descends 25 feet per second.

1.) What is an equation that models the altitude of the airplane?

2.) What is the altitude of the plane after 30 seconds?

Linear Growth:

bxay


Amount of change (Slope)

xy 255000

30255000 y

4250y

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Exponential Growth

Exponential growth is similar to linear growth except exponential growth grows by multiplying each period rather than adding.

Exponential Growth:

xaby


Percent of change (Growth factor)

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Graphs of Exponential Growth

Here is a graph of a simple exponential growth equation.

xy 32


Every period the amount triples.

When x = 0, y = 2.

The graph gets steeper and steeper as x gets bigger.

The graph gets flatter and flatter as x gets more negative

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Graphs of Exponential Growth

Here is a graph of a simple exponential growth equation.

xy 5.8


When x = 0, y = 8.

The graph gets steeper and steeper as x gets more negative.

The graph gets fatter and flatter as x gets bigger

Every period the amount gets cut in half.

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Growth FactorsWhat does the graph of the following function look like?

xy 13Why does the graph stay flat? Why doesn’t it go up or down?

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What if the growth factor is greater than one? This is the graph of:

y = 3 (1.1)x

If the growth factor is greater than one, the graph goes up. This is called exponential growth.

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What if the growth factor is less than one? This is the graph of:

y = 3 (1.1)x

This is the graph of:

y = 3(0.9)x

If the growth factor is less than one, the graph goes down. This is called exponential decay.

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Growth Factors

GrowthWhen the growth factor is greater than one, the equation represents growth.

The amount of growth is equal to the difference between the growth factor and one.

DecayWhen the growth factor is less than one, the equation represents decay.

The amount of decay is equal to the difference between the growth factor and one.

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What is the growth factor?

10% growth

Growth factor = 1.10

25% growth


20.9% growth


10% decay


25% decay


20.9% decay


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How much growth or decay is represented by this growth factor?

30% growth

Growth factor = 1.3

85% growth


42.5% growth


80% decay


5% decayGrowth factor = 0.95

60.7% decayGrowth factor = 0.393

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What is the growth factor?

100% growth

Growth factor = 2

525% growth


237.85% growth


350% decay

Decay cannot

exceed 100%. Why?

Because you cannot lose more than you start with.

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How much growth or decay is represented by this growth factor?

130% growth

Growth factor = 2.3

328% growth


900% growth

Growth factor = 10

99% decay




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An ant colony had ten ants on March 1st. Its population has been growing

65% every week.

• What equation models the population of ants as it grows from week to week?

y 10(1.65)x

• By the end of April, eight weeks have gone by, how many ants are now in the colony?

549.378 (549 ants)

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Some doctors are testing the effectiveness of a new antibiotic. At 2:00 PM, they place the antibiotic in a Petri dish with 18 grams of

bacteria. The antibiotic kills the bacteria at a rate of 12% per hour.

• What equation models the amount of bacteria from hour to hour?

y 18(0.88)x

At 7:00 PM, how much bacteria remains in the dish?

9.499 grams

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Finding an exponential equation that passes through points.

• Find the exponential equation that passes through: (0,12) & (3,1.5)

y a b x Pick an easy point to fill in for x & y.

12 a b 0b0 = 1, so a = 12

y 12 b x Plug in the other point and solve for b.

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Find the exponential equation that passes through: (0,12) & (3,1.5)

y 12 b x Plug in the other point and solve for b.

1.5 12 b 3

1.5

12b3

1

8b3

b1

2

So the equation is:

y 121

2

x

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• Find the exponential equation that passes through: (2,20) & (6,320)

y a b x In this case, plugging in doesn’t eliminate either a or b.

320 a b 6

20 a b 2

Plug in both separately and then stack them.

Divide to eliminate a.

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Find the exponential equation that passes through: (2,20) & (6,320)

320

20a b 6

a b 2Simplify these fractions

16 b4

b2Solve for b.

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Find the exponential equation that passes through: (2,20) & (6,320)

y a 2 x Plug in b to solve for a.

20 a 2 2

20 4a

a5

So the equation is:

y 5 2 x

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Find the equation of the exponential that passes through (0,10) & (2,16)

xy 265.110

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xy 889.0125.10

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xy 23

25


xy 094.1467.4

1 8.1 exponential growth 2 a review of linear growth in algebra 1, almost every real life situation...

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