Company kind of A has 100 stores, and broadens by opening 50 new stores a yearCompany B has 100 stores, and increases by increasing the variety of stores by 50% of their total annually.

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Company type of A is exhibiting straight development. In direct expansion, we have a continuous rate of change—a consistent number that the output enhanced for each increase in input. For company A, the number of new stores per year is the very same yearly.

Company kind of B is different—we have a percent price of change fairly than a continuous number of stores/year as our price of readjust. To see the meaning of this difference compare a 50% increase as soon as tbelow are 100 stores to a 50% boost when tright here are 1000 stores:

100 stores, a 50% boost is 50 stores in that year.1000 stores, a 50% increase is 500 stores in that year.

Calculating the number of stores after several years, we can plainly watch the distinction in results.

YearsCompany ACompany type of B


Graphs of data from A and B, via B fit to a curve.

This percent growth have the right to be modeled via an exponential feature.

Exponential Function

Anexponential growth or degeneration function is a function that grows or shrinks at a continuous percent growth rate. The equation can be composed in the develop f(x) = a(1 + r)x or f(x) = abx wright here b = 1 + r.


a is the initial or starting value of the feature,r is the percent development or decay rate, created as a decimal,b is the development variable or growth multiplier. Due to the fact that powers of negative numbers behave actually strangely, we limit b to positive values.

Shana Calamethod, Dale Hoffguy, and also David Lippmale, Firm Calculus, ”1.7: Exponential Functions,” licensed under a CC-BY license.

Graphical Features of Exponential Functions

Graphically, in the functionf(x) = abx.

a is the vertical intercept of the graph.b determines the price at which the graph grows:the feature will certainly rise if b > 1,the function will certainly decrease if 0 The graph will have a horizontal asymptote at y = 0.The graph will certainly be concave up if a>0; concave down ifa The doprimary of the function is all actual numbers.The range of the attribute is (0,∞) if a > 0, and also (−∞,0) if a

When sketching the graph of an exponential function, it have the right to be helpful to remember that the graph will certainly pass through the points (0,a) and also (1, ab).

The value bwill recognize the function’s lengthy run behavior:

If b > 1, as x → ∞, f(x) → ∞, and as x → −∞, f(x) → 0.If 0 ) → 0, and as x → –∞, f(x) → ∞.Example 6

Lay out a graph ofdisplaystylef(x)=4(frac13)^x

This graph will have actually a vertical intercept at (0,4), and also pass through the pointdisplaystyle(1,frac43). Due to the fact that b 0, the graph will be concave up.


We can also watch from the graph the long run behavior: asx → ∞ , f(x) → 0, and as x → –∞, f(x) → ∞.

To acquire a much better feeling for the result ofa and b on the graph, study the sets of graphs listed below. The first set mirrors miscellaneous graphs, wright here a stays the same and also we only change the worth for b. Notice that the closer the worth of is to 1, the much less steep the graph will be.


Changing the worth ofb.

In the next collection of graphs,is altered and our value for b stays the same.


Changing the value of a.

Notice that changing the value for a transforms the vertical intercept. Since is multiplying thebx term, a acts as a vertical stretch variable, not as a change. Notice also that the lengthy run habits for every one of these functions is the very same because the expansion variable did not change and also namong these worths introduced a vertical flip.

Try it for yourself usingthis applet.

Example 7

Match each equation via its graph.



The graph ofk(x) is the simplest to determine, given that it is the only equation with a development element less than one, which will develop a decreasing graph. The graph of h(x) can be figured out as the only flourishing exponential attribute via a vertical intercept at (0,4). The graphs of f(x) and g(x) both have actually a vertical intercept at (0,2), however considering that g(x) has actually a bigger expansion element, we have the right to recognize it as the graph increasing faster.


Shana Calamethod, Dale Hoffguy, and also David Lippguy, Company Calculus, ”1.7: Exponential Functions,” licensed under a CC-BY license.

Applications of Exponential Functions

Example 1

India’s populace was 1.14 billion in the year 2008 and also is thriving by around 1.34% annually. Write an exponential function for India’s population, and also use it to predict the populace in 2020.

Using 2008 as our founding time (), our initial population will be 1.14 billion. Due to the fact that the percent development rate was 1.34%, our value for is 0.0134.

Using the fundamental formula for exponential growthwe deserve to create the formula,f(t) = 1.14(1 + 0.0134)t

To estimate the population in 2020, we evaluate the feature at, because 2020 is 12 years after 2008:

Example 2

A certificate of deposit (CD) is a type of savings account readily available by banks, frequently offering a higher interemainder price in rerevolve for a solved size of time you will leave your money invested. If a financial institution provides a 24 month CD via an annual interemainder rate of 1.2% compounded monthly, exactly how a lot will a $1000 investment grow to over those 24 months?

First, we need to notification that the interemainder price is an yearly price, however is compounded monthly, meaning interemainder is calculated and also added to the account monthly. To uncover the monthly interest price, we divide the yearly price of 1.2% by 12 because tbelow are 12 months in a year: 1.2%/12 = 0.1%. Each month we will certainly earn 0.1% interemainder. From this, we deserve to erected an exponential feature, with our initial amount of $1000 and a development price of, and also our input measured in months:


After 24 months, the account will certainly have actually grown to.

Example 3

Bismuth-210 is an isotope that radioproactively decays by around 13% each day, definition 13% of the continuing to be Bismuth-210 transcreates right into one more atom (polonium-210 in this case) each day. If you start via 100 mg of Bismuth-210, how much remains after one week?

With radioactive degeneration, rather of the quantity increasing at a percent price, the amount is decreasing at a percent price. Our initial quantity ismg, and our expansion rate will be negative 13%, considering that we are decreasing: . This provides the equation

This have the right to also be explained by recognizing that if 13% decays, then 87% stays.

After one week, 7 days, the amount remaining would bemg of Bismuth-210 stays.

Euler’s Number: e

Due to the fact that e is often supplied as the base of an exponential, the majority of scientific and graphing calculators have a button that deserve to calculate powers of e, usually labeled exp(x). Some computer software application rather defines a duty exp(x), where exp(x) = . Since calculus studies constant change, we will certainly almost always use the -based develop of exponential equations in this course.

Continuous Growth Formula

Continuous growth deserve to be calculated making use of the formula where

is the beginning amount,is the constant growth price.Example 4

Radon-222 decays at a continuous rate of 17.3% per day. How a lot will certainly 100mg of Radon-222 degeneration to in 3 days?

Due to the fact that we are provided a continuous decay rate, we usage the consistent expansion formula. Because the substance is decaying, we understand the expansion price will be negative:, mg of Radon-222 will remajor.

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Shana Calameans, Dale Hoffguy, and also David Lippmale, Company Calculus, ”1.7: Exponential Functions,” licensed under a CC-BY license.