In addition to linear, quadratic, rational, and radical functions, there room A function of the type f(x) = bx, where b > 0 and b ≠ 1.

You are watching: Which function represents a vertical stretch of an exponential function?


")">exponential functions
. Exponential attributes have the type f(x) = bx, wherein b > 0 and also b ≠ 1. Just as in any kind of exponential expression, b is dubbed the The expression the is being raised to a power once using exponential notation. In 53, 5 is the base, i beg your pardon is the number the is consistently multiplied. 53 = 5 • 5 • 5. In ab, a is the base.


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and also x is called the When a number is to express in the kind ab, b is the exponent. The exponent suggests how plenty of times the base is used as a factor. Power and exponent median the exact same thing.


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.

An example of one exponential duty is the growth of bacteria. Part bacteria dual every hour. If you begin with 1 bacterium and also it doubles every hour, girlfriend will have actually 2x bacteria after x hours. This have the right to be created as f(x) = 2x.

Before girlfriend start, f(0) = 20 = 1

After 1 hour f(1) = 21 = 2

In 2 hrs f(2) = 22 = 4

In 3 hours f(3) = 23 = 8

and therefore on.

With the meaning f(x) = bx and the limitations that b > 0 and that b ≠ 1, the domain of an exponential role is the collection of all real numbers. The variety is the collection of all hopeful real numbers. The complying with graph mirrors f(x) = 2x.

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Exponential Growth


As you deserve to see above, this exponential role has a graph the gets an extremely close to the x-axis as the graph extends to the left (as x becomes an ext negative), yet never really touches the x-axis. Understanding the basic shape the the graphs that exponential attributes is useful for graphing certain exponential equations or functions.

Making a table of values is likewise helpful, due to the fact that you can use the table to location the curve of the graph more accurately. One point to mental is that if a base has a negative exponent, climate take the reciprocal of the base to do the exponent positive. Because that example,

*
.


Example

Problem

Make a table of worths for f(x) = 3x.

x

f(x)

Make a “T” to start the table v two columns. Label the columns x and f(x).

x

f(x)

−2

−1

0

1

2

Choose numerous values for x and put them as separate rows in the x column.

Tip: it’s always an excellent to include 0, optimistic values, and negative values, if you can.

Answer

x

f(x)

−2

−1

0

1

1

3

2

9

Evaluate the role for each worth of x, and write the an outcome in the f(x) column next to the x worth you used. For example, as soon as

x = −2, f(x) = 3-2 =

*
 = , for this reason  goes in the f(x) column alongside −2 in the x column. F(1) = 31 = 3, so 3 walk in the f(x) column beside 1 in the x column.

Note that your table of values might be various from who else’s, if you decided different numbers because that x.


Look at the table of values. Think about what happens together the x values increase—so carry out the duty values (f(x) or y)!

Now the you have a table that values, you have the right to use these worths to aid you draw both the shape and also location of the function. Connect the point out as ideal you can to make a smooth curve (not a collection of straight lines). This mirrors that every one of the clues on the curve are part of this function.


Example

Problem

Graph f(x) = 3x.

x

f(x)

−2

−1

0

1

1

3

2

9

Start through a table that values, like the one in the example above.

x

f(x)

point

−2

(−2, )

−1

(−1, )

0

1

(0, 1)

1

3

(1, 3)

2

9

(2, 9)

If friend think that f(x) together y, each row creates an bespeak pair the you deserve to plot top top a coordinate grid.

*

Plot the points.

Answer

*

Connect the points as finest you can, making use of a smooth curve (not a collection of straight lines). Use the form of one exponential graph to assist you: this graph gets an extremely close to the

x- axis ~ above the left, yet never really touches the x-axis, and also gets steeper and steeper top top the right.


This is an example of An exponential role of the type f(x) = bx, where b > 1, and also b ≠ 1. The function increases as x increases.


")">exponential growth
. Together x increases, f(x) “grows” much more quickly. Let’s try another one.


Example

Problem

Graph f(x) = 4x.


x

f(x)

−2

−1

0

1

1

4

2

16


Start v a table of values. You can pick different values, yet once again, it’s useful to incorporate 0, some optimistic values, and some negative values.

Remember,

4-2 =

*
 = .

If girlfriend think that f(x) as y, every row forms an bespeak pair the you deserve to plot ~ above a coordinate grid.

*

Plot the points.

Notice the the bigger base in this difficulty made the duty value skyrocket. Even with x as tiny as 2, the role value is too huge for the axis range you used before. Friend can adjust the scale, however then our other values are an extremely close together. You might also shot other points, such as when x =

*
. Because you recognize the square source of 4, friend can find that worth in this case:
*
. The point
*
 is the blue allude on this graph.

For other bases, you could need to use a calculator to assist you uncover the duty value.

 

Answer

*

Connect the clues as best you can, using a smooth curve (not a collection of directly lines). Usage the shape of one exponential graph to aid you: this graph gets an extremely close come the x-axis on the left, but never really touches the

x- axis, and gets steeper and also steeper on the right.


Let’s to compare the three graphs did you do it seen. The functions f(x) = 2x, f(x) = 3x, and also

f(x) = 4x are all graphed below.

*

Notice the a bigger base makes the graph steeper. A bigger base additionally makes the graph closer come the y-axis because that x > 0 and closer to the x-axis because that x


Exponential Decay


Remember that for exponential functions, b > 0, but b ≠ 1. In the instances above, b > 1. What happens when b is in between 0 and also 1, 0 b


Example

Problem

Graph

*
.


x

f(x)

−2

4

−1

2

0

1

1

*

2


Start with a table that values.

Be cautious with the negative exponents! remember to take it the reciprocal of the base to make the exponent positive. In this case,

*
, and also
*
.

*

Use the table together ordered pairs and also plot the points.

Answer

*

Since the points space not ~ above a line, friend can’t use a straightedge. Affix the clues as finest you deserve to using a smooth curve (not a collection of directly lines).


Notice the the form is similar to the shape once b > 1, however this time the graph gets closer to the x-axis as soon as x > 0, fairly than once x an exponential function of the kind f(x) = bx, whereby 0 b . The role decreases together x increases.


")">exponential decay. Instead of the duty values “growing” together x worths increase, as they walk before, the function values “decay” or decrease together x worths increase. They obtain closer and closer come 0.


Example

Problem

Graph

*
.


x

f(x)

−2

16

-1

4

0

1

1

2


Create a table that values. Again, be careful with the an unfavorable exponents. Psychic to take it the reciprocal of the basic to make the exponent positive.

*
.

Notice that in this table, the x worths increase. The y values decrease.

*

Use the table pairs to plot points. You may want come include brand-new points, specifically when one of the points from the table, right here (−2, 16) i will not ~ fit on her graph. Since you know the square root of 4, shot x =. You can uncover that value in this case:

*
.

The allude (, 8) has been included in blue. You might feel it essential to include added points. You additionally may must use a calculator, relying on the base.

Answer

*

Connect the points as best you can, using a smooth curve.


Which of the following is a graph because that

*
?

A)

B)

C)

D)

*


Show/Hide Answer

A)

 

Incorrect. This graph is increasing, since the f(x) or y values boost as the x values increase. (Compare the values for x = 1 and also x = 2.) This graph mirrors exponential growth, with a base higher than 1. The exactly answer is Graph D.

B)

 

Incorrect. This graph is decreasing, yet all the function values room negative. The selection for one exponential duty is constantly positive values. The exactly answer is Graph D.

C)

 

Incorrect. This graph is increasing, however all the duty values are negative. The exactly graph should be decreasing v positive role values. The correct answer is Graph D.

D)

 

*

Correct. All the duty values room positive, and the graph is decreasing (showing exponential decay).

Applying Exponential Functions


Exponential functions can be provided in numerous contexts, such together compound attention (money), populace growth, and radioactive decay. In many of these, however, the role is not precisely of the form f(x) = bx. Often, this is changed by adding or multiplying constants.

For example, the compound interest formula is

*
, wherein P is the principal (the initial invest that is gathering interest) and also A is the amount of money you would certainly have, v interest, in ~ the finish of t years, making use of an yearly interest price of r (expressed together a decimal) and m compounding durations per year. In this case the base is the value stood for by the expression 1 +
*
 and the exponent is mt—a product of 2 values.


Example

Problem

If girlfriend invest $1,000 in one account payment 4% interest, compounded quarterly, how much money will certainly you have actually after 3 years?

The money girlfriend will have after 3 years will be A.

P = $1,000

r = 0.04

m = 4

t = 3

First recognize which of A, P, r, m, and t is being asked for, then determine values for the continuing to be variables.

The primary is $1,000.

The price is 4% = 0.04.

The time in years is 3.

Compounded quarterly means 4 times a year.

*

To uncover the lot A, usage the formula.

Answer

You will have $1,126.83 after ~ 3 years.

Round the number come the nearest cent (hundredth).

Notice the this means the quantity of interest earned after three years is $126.83. ($1,126.83, minus the principal, $1,000).


Radioactive degeneration is an example of exponential decay. Radioactive facets have a half-life. This is the lot of time it takes for half of a massive of the element to degeneration into another substance. For example, uranium-238 is a slowly decaying radioactive aspect with a half-life of about 4.47 billion years. That way it will take that lengthy for 100 grams that uranium-238 come turn right into 50 grams the uranium-238 (the various other 50 grams will have turned into one more element). It is a lengthy time! ~ above the other extreme, radon-220 has actually a half-life of about 56 seconds. What go this mean? 100 grams that radon-220 will turn into 50 grams the radon-220 and also 50 grams the something rather in less than a minute!

Since the amount is halved each half-life, one exponential role can be offered to describe the amount remaining over time. The formula

*
 gives the remaining amount R from an initial amount A, whereby h is the half-life of the element and also t is the amount of time passed (using the exact same time unit together the half-life).


Example

Problem

Caesium-137 is a radioactive aspect used in clinical applications. It has a half-life of around 30 years. Intend a laboratory has actually 10 grams that caesium-137. If they don’t use it, exactly how much will certainly still be caesium-137 in 60 years?

R: This is the continuing to be value, what you room trying to find.

A: The initial amount to be 10 grams.

h: The half-life is 30 years.

t: The amount of time passed is 60 years. (Note that this is in the very same unit, years, as the half-life.)

Identify the values well-known in the formula.

*

Use the formula.

Answer

There will certainly be 2.5 grams that caesium-137 in 60 years.


Billy’s mom put $100 into a bank account because that him once he was born. The account got interest at a price of 3% per year, compounded monthly. Presume no more money to be deposited and none was withdrawn, how much money will be in the account when Billy transforms 18?

A) $170.24

B) $171.49

C) $8561.76

D) $20,718.34


Show/Hide Answer

A) $170.24

Incorrect. This is the amount as soon as the yearly rate is compounded yearly (m = 1). In this case we space compounding monthly, m = 12. The correct answer is $171.49.

B) $171.49

Correct. Making use of a rate r of 0.03 and 12 compounding durations per year (m = 12), the formula offers $171.49.

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C) $8561.76

Incorrect. Girlfriend may have actually used r = 3 quite than r = 0.03 for the annual rate, then misread the an outcome from her calculator. The exactly answer is $171.49.

D) $20,718.34

Incorrect. Friend may have used r = 0.3 fairly than r = 0.03 because that the annual rate. The exactly answer is $171.49.

Summary


Exponential attributes of the kind f(x) = bx show up in different contexts, including finance and also radioactive decay. The basic b should be a optimistic number and cannot be 1. The graphs that these attributes are curve that increase (from left to right) if b > 1, mirroring exponential growth, and decrease if 0 b