In enhancement to linear, quadratic, rational, and radical features, tright here are A feature of the develop f(x) = bx, where b > 0 and b ≠ 1.

You are watching: Which is a stretch of an exponential decay function?


")">exponential functions
. Exponential features have the form f(x) = bx, where b > 0 and also b ≠ 1. Just as in any exponential expression, b is dubbed the The expression that is being increased to a power when making use of exponential notation. In 53, 5 is the base, which is the number that is continuously multiplied. 53 = 5 • 5 • 5. In ab, a is the base.


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and x is called the When a number is expressed in the form ab, b is the exponent. The exponent indicates how many kind of times the base is offered as a element. Power and also exponent suppose the very same point.


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.

An instance of an exponential feature is the growth of bacteria. Some bacteria twin eincredibly hour. If you start via 1 bacterium and it doubles every hour, you will certainly have actually 2x bacteria after x hrs. This have the right to be created as f(x) = 2x.

Before you begin, f(0) = 20 = 1

After 1 hour f(1) = 21 = 2

In 2 hours f(2) = 22 = 4

In 3 hrs f(3) = 23 = 8

and so on.

With the definition f(x) = bx and also the restrictions that b > 0 and also that b ≠ 1, the domain of an exponential feature is the set of all real numbers. The range is the set of all positive genuine numbers. The complying with graph reflects f(x) = 2x.

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Exponential Growth


As you have the right to see over, this exponential attribute has actually a graph that gets very close to the x-axis as the graph exhas a tendency to the left (as x becomes more negative), yet never really touches the x-axis. Knowing the basic shape of the graphs of exponential functions is helpful for graphing certain exponential equations or attributes.

Making a table of values is also beneficial, bereason you deserve to usage the table to place the curve of the graph even more accurately. One thing to remember is that if a base has actually a negative exponent, then take the reciprocal of the base to make the exponent positive. For instance,

*
.


Example

Problem

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

x

f(x)

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

x

f(x)

−2

−1

0

1

2

Choose a number of worths for x and put them as separate rows in the x column.

Tip: It’s always great to encompass 0, positive values, and negative values, if you deserve to.

Answer

x

f(x)

−2

−1

0

1

1

3

2

9

Evaluate the function for each worth of x, and create the result in the f(x) column alongside the x value you offered. For instance, when

x = −2, f(x) = 3-2 =

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

Keep in mind that your table of worths may be different from someone else’s, if you decided various numbers for x.


Look at the table of values. Think around what happens as the x values increase—so do the feature worths (f(x) or y)!

Now that you have a table of worths, you deserve to use these worths to aid you draw both the form and place of the function. Connect the points as finest you have the right to to make a smooth curve (not a collection of right lines). This mirrors that every one of the points on the curve are part of this attribute.


Example

Problem

Graph f(x) = 3x.

x

f(x)

−2

−1

0

1

1

3

2

9

Start via a table of values, prefer the one in the instance above.

x

f(x)

point

−2

(−2, )

−1

(−1, )

0

1

(0, 1)

1

3

(1, 3)

2

9

(2, 9)

If you think of f(x) as y, each row develops an ordered pair that you can plot on a coordinate grid.

*

Plot the points.

Answer

*

Connect the points as ideal you deserve to, making use of a smooth curve (not a series of right lines). Use the shape of an exponential graph to aid you: this graph gets very cshed to the

x- axis on the left, but never before really touches the x-axis, and gets steeper and steeper on the best.


This is an example of An exponential attribute of the create f(x) = bx, where b > 1, and b ≠ 1. The function boosts as x boosts.


")">exponential growth
. As x boosts, f(x) “grows” even more quickly. Let’s attempt an additional one.


Example

Problem

Graph f(x) = 4x.


x

f(x)

−2

−1

0

1

1

4

2

16


Start through a table of worths. You have the right to select various values, but as soon as aget, it’s helpful to include 0, some positive worths, and some negative worths.

Remember,

4-2 =

*
 = .

If you think of f(x) as y, each row develops an ordered pair that you have the right to plot on a coordinate grid.

*

Plot the points.

Notice that the larger base in this trouble made the function worth skyrocket. Even with x as small as 2, the feature value is also big for the axis range you provided before. You can readjust the range, however then our various other worths are exceptionally close together. You could additionally try various other points, such as once x =

*
. Since you understand the square root of 4, you have the right to discover that value in this case:
*
. The suggest
*
 is the blue point on this graph.

For other bases, you can should usage a calculator to assist you uncover the feature value.

 

Answer

*

Connect the points as ideal you can, making use of a smooth curve (not a series of right lines). Use the form of an exponential graph to aid you: this graph gets exceptionally close to the x-axis on the left, however never really touches the

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


Let’s compare the three graphs you’ve seen. The attributes f(x) = 2x, f(x) = 3x, and

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

*

Notice that a bigger base makes the graph steeper. A larger base also provides the graph closer to the y-axis for x > 0 and closer to the x-axis for x


Exponential Decay


Remember that for exponential features, b > 0, but b ≠ 1. In the examples over, b > 1. What happens once b is between 0 and 1, 0 b


Example

Problem

Graph

*
.


x

f(x)

−2

4

−1

2

0

1

1

*

2


Start through a table of values.

Be cautious via the negative exponents! Remember to take the reciprocal of the base to make the exponent positive. In this situation,

*
, and
*
.

*

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

Answer

*

Since the points are not on a line, you can’t use a straightedge. Connect the points as finest you have the right to using a smooth curve (not a series of directly lines).


Notice that the form is equivalent to the shape once b > 1, but this time the graph gets closer to the x-axis once x > 0, fairly than as soon as x An exponential attribute of the create f(x) = bx, where 0 b . The feature decreases as x increases.


")">exponential decay. Instead of the function worths “growing” as x worths boost, as they did prior to, the feature worths “decay” or decrease as x values increase. They obtain closer and also closer to 0.


Example

Problem

Graph

*
.


x

f(x)

−2

16

-1

4

0

1

1

2


Create a table of worths. Aget, be mindful through the negative exponents. Remember to take the reciprocal of the base to make the exponent positive.

*
.

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

*

Use the table pairs to plot points. You may desire to encompass new points, specifically when among the points from the table, below (−2, 16) won’t fit on your graph. Due to the fact that you understand the square root of 4, attempt x =. You deserve to find that value in this case:

*
.

The suggest (, 8) has actually been included in blue. You might feel it crucial to encompass additional points. You also may need to use a calculator, relying on the base.

Answer

*

Connect the points as finest you have the right to, making use of a smooth curve.


Which of the adhering to is a graph for

*
?

A)

B)

C)

D)

*


Show/Hide Answer

A)

 

Incorrect. This graph is raising, bereason the f(x) or y worths increase as the x values rise. (Compare the worths for x = 1 and x = 2.) This graph shows exponential growth, with a base greater than 1. The correct answer is Graph D.

B)

 

Incorrect. This graph is decreasing, however all the feature values are negative. The range for an exponential function is constantly positive values. The correct answer is Graph D.

C)

 

Incorrect. This graph is boosting, but all the attribute worths are negative. The correct graph need to be decreasing through positive feature worths. The correct answer is Graph D.

D)

 

*

Correct. All the function values are positive, and also the graph is decreasing (mirroring exponential decay).

Applying Exponential Functions


Exponential attributes deserve to be used in many kind of conmessages, such as compound interemainder (money), population development, and also radioenergetic degeneration. In most of these, however, the attribute is not precisely of the create f(x) = bx. Often, this is readjusted by adding or multiplying constants.

For example, the compound interest formula is

*
, wbelow P is the principal (the initial investment that is gathering interest) and also A is the amount of money you would certainly have actually, via interest, at the finish of t years, making use of an annual interemainder price of r (expressed as a decimal) and also m compounding durations per year. In this situation the base is the worth represented by the expression 1 +
*
 and the exponent is mt—a product of 2 values.


Example

Problem

If you invest $1,000 in an account paying 4% interemainder, compounded quarterly, just how much money will you have actually after 3 years?

The money you will have actually after 3 years will certainly be A.

P = $1,000

r = 0.04

m = 4

t = 3

First determine which of A, P, r, m, and also t is being asked for, then determine worths for the remaining variables.

The principal is $1,000.

The rate is 4% = 0.04.

The time in years is 3.

Compounded quarterly indicates 4 times a year.

*

To uncover the amount A, usage the formula.

Answer

You will certainly have actually $1,126.83 after 3 years.

Round the number to the nearest cent (hundredth).

Notice that this means the amount of interest earned after 3 years is $126.83. ($1,126.83, minus the major, $1,000).


Radioactive degeneration is an example of exponential decay. Radioenergetic aspects have actually a half-life. This is the amount of time it takes for half of a mass of the facet to degeneration right into another substance. For example, uranium-238 is a progressively decaying radioenergetic aspect with a half-life of around 4.47 billion years. That indicates it will certainly take that long for 100 grams of uranium-238 to turn right into 50 grams of uranium-238 (the other 50 grams will certainly have turned right into an additional element). That’s a long time! On the various other too much, radon-220 has a half-life of around 56 seconds. What does this mean? 100 grams of radon-220 will turn into 50 grams of radon-220 and 50 grams of something else in much less than a minute!

Due to the fact that the amount is halved each half-life, an exponential function have the right to be offered to describe the amount remaining over time. The formula

*
 provides the staying amount R from an initial amount A, where h is the half-life of the element and also t is the amount of time passed (making use of the very same time unit as the half-life).


Example

Problem

Caesium-137 is a radioenergetic facet supplied in medical applications. It has actually a half-life of around 30 years. Suppose a laboratory has actually 10 grams of caesium-137. If they don’t use it, exactly how a lot will certainly still be caesium-137 in 60 years?

R: This is the continuing to be worth, what you are trying to uncover.

A: The initial amount was 10 grams.

h: The half-life is 30 years.

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

Identify the worths recognized in the formula.

*

Use the formula.

Answer

Tbelow will certainly be 2.5 grams of caesium-137 in 60 years.


Billy’s mommy put $100 right into a bank account for him as soon as he was born. The account acquired interest at a price of 3% per year, compounded monthly. Assuming no more money was deposited and also none was withdrawn, exactly how a lot money will certainly 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 when the yearly rate is compounded each year (m = 1). In this situation we are compounding monthly, m = 12. The correct answer is $171.49.

B) $171.49

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

See more: Solved: How Many Unpaired Electrons Does Phosphorus Have In Its Valence Shell?

C) $8561.76

Incorrect. You might have provided r = 3 fairly than r = 0.03 for the annual price, then misreview the outcome from your calculator. The correct answer is $171.49.

D) $20,718.34

Incorrect. You may have supplied r = 0.3 rather than r = 0.03 for the annual price. The correct answer is $171.49.

Summary


Exponential attributes of the create f(x) = bx show up in different conmessages, including finance and also radioenergetic degeneration. The base b need to be a positive number and cannot be 1. The graphs of these features are curves that boost (from left to right) if b > 1, mirroring exponential development, and also decrease if 0 b