In enhancement to direct, quadratic, rational, and also radical functions, tbelow are A feature of the form f(x) = bx, where b > 0 and b ≠ 1.

You are watching: Why do exponential functions grow faster than polynomials

")">exponential functions
. Exponential attributes have the create f(x) = bx, wright here b > 0 and b ≠ 1. Just as in any exponential expression, b is called the The expression that is being elevated to a power as soon as 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.

")">base
and also x is dubbed the When a number is expressed in the form ab, b is the exponent. The exponent shows just how many type of times the base is offered as a variable. Power and also exponent suppose the very same thing.

")">exponent
.

An example of an exponential attribute is the expansion of bacteria. Some bacteria double eextremely hour. If you begin via 1 bacterium and it doubles every hour, you will have 2x bacteria after x hrs. This can be created as f(x) = 2x.

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

After 1 hour f(1) = 21 = 2

In 2 hours f(2) = 22 = 4

In 3 hours f(3) = 23 = 8

and so on.

With the interpretation f(x) = bx and the restrictions that b > 0 and also that b ≠ 1, the doprimary of an exponential function is the collection of all actual numbers. The range is the collection of all positive actual numbers. The following graph mirrors f(x) = 2x.

Exponential Growth

As you deserve to view over, this exponential function has a graph that gets extremely close to the x-axis as the graph extends to the left (as x becomes more negative), however never before really touches the x-axis. Knowing the general shape of the graphs of exponential attributes is useful for graphing particular exponential equations or functions.

Making a table of worths is also beneficial, bereason you can use the table to area the curve of the graph even more accurately. One point 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 example,

.

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

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

x

 f(x) −2 −1 0 1 2

Choose numerous worths for x and put them as sepaprice rows in the x column.

Tip: It’s always excellent to encompass 0, positive values, and also negative values, if you have the right to.

x

 f(x) −2 −1 0 1 1 3 2 9

Evaluate the feature for each value of x, and compose the cause the f(x) column next to the x value you used. For example, when

x = −2, f(x) = 3-2 =

= , so  goes in the f(x) column next to −2 in the x column. f(1) = 31 = 3, so 3 goes in the f(x) column next to 1 in the x column.

Note that your table of values may be different from someone else’s, if you made a decision various numbers for x.

Look at the table of values. Think about what happens as the x worths increase—so perform the function values (f(x) or y)!

Now that you have actually a table of worths, you deserve to use these values to assist you attract both the shape and also area of the attribute. Connect the points as best you deserve to to make a smooth curve (not a series of right lines). This mirrors that every one of the points on the curve are component 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 creates an ordered pair that you can plot on a coordinate grid.

Plot the points.

Connect the points as ideal you can, using a smooth curve (not a series of directly lines). Use the form of an exponential graph to assist you: this graph gets exceptionally cshed to the

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

This is an instance of An exponential attribute of the form f(x) = bx, wright here b > 1, and also b ≠ 1. The function increases as x rises.

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

 Example Problem Graph f(x) = 4x. x f(x) −2 −1 0 1 1 4 2 16

Start with a table of values. You have the right to select different worths, yet as soon as aobtain, it’s useful to incorporate 0, some positive values, and some negative values.

Remember,

4-2 =

= .

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.

Notice that the larger base in this difficulty made the function value skyrocket. Even with x as little as 2, the function value is also huge for the axis scale you used before. You have the right to adjust the scale, yet then our other worths are exceptionally cshed together. You might additionally attempt various other points, such as once x =

. Due to the fact that you know the square root of 4, you deserve to discover that value in this case:
. The suggest
is the blue allude on this graph.

For various other bases, you could have to usage a calculator to aid you uncover the feature value.

Connect the points as ideal you can, utilizing a smooth curve (not a series of directly lines). Use the shape of an exponential graph to help you: this graph gets exceptionally cshed to the x-axis on the left, however never really touches the

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

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

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

Notice that a larger base renders the graph steeper. A bigger 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 attributes, b > 0, but b ≠ 1. In the examples over, b > 1. What happens when b is in 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 case,

, and
.

Use the table as ordered pairs and plot the points.

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

Notice that the form is similar to the shape as soon as b > 1, yet this time the graph gets closer to the x-axis once x > 0, rather than as soon as x An exponential feature of the develop f(x) = bx, wbelow 0 b . The feature decreases as x boosts.

")">exponential decay. Instead of the attribute values “growing” as x values boost, as they did prior to, the feature worths “decay” or decrease as x worths rise. They acquire 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. Again, 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 boost. The y values decrease.

Use the table pairs to plot points. You may want to incorporate new points, specifically once one of the points from the table, right here (−2, 16) won’t fit on your graph. Since you recognize the square root of 4, try x =. You can uncover that value in this case:

.

The suggest (, 8) has actually been consisted of in blue. You might feel it important to encompass extra points. You likewise may should use a calculator, depending on the base.

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

Which of the complying with is a graph for

?

A)

B)

C)

D)

A)

Incorrect. This graph is enhancing, because the f(x) or y values increase as the x values boost. (Compare the values for x = 1 and also x = 2.) This graph reflects exponential expansion, via a base higher than 1. The correct answer is Graph D.

B)

Incorrect. This graph is decreasing, yet all the function worths are negative. The selection for an exponential function is always positive values. The correct answer is Graph D.

C)

Incorrect. This graph is boosting, however all the attribute worths are negative. The correct graph must be decreasing through positive feature values. The correct answer is Graph D.

D)

Correct. All the attribute worths are positive, and also the graph is decreasing (showing exponential decay).

Applying Exponential Functions

Exponential attributes deserve to be offered in many conmessages, such as compound interest (money), population growth, and also radioenergetic decay. In many of these, yet, the attribute is not precisely of the form 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 A is the amount of money you would have, via interemainder, at the end of t years, making use of an yearly interemainder price of r (expressed as a decimal) and also m compounding periods per year. In this situation the base is the value represented by the expression 1 +
and also the exponent is mt—a product of 2 worths.

 Example Problem If you invest \$1,000 in an account paying 4% interest, compounded quarterly, how much money will certainly you have 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 recognize which of A, P, r, m, and also t is being asked for, then recognize values for the remaining variables. The primary is \$1,000. The rate is 4% = 0.04. The time in years is 3. Compounded quarterly suggests 4 times a year. To discover the amount A, use the formula. Answer You will certainly have actually \$1,126.83 after 3 years. Round the number to the nearemainder cent (hundredth). Notice that this implies the amount of interemainder earned after 3 years is \$126.83. (\$1,126.83, minus the principal, \$1,000).

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

Since the amount is halved each half-life, an exponential attribute can be used to explain the amount remaining over time. The formula

gives the remaining amount R from an initial amount A, wright here h is the half-life of the aspect and also t is the amount of time passed (utilizing the same time unit as the half-life).

 Example Problem Caesium-137 is a radioenergetic element offered 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, how much will certainly still be caesium-137 in 60 years? R: This is the remaining value, 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. (Note that this is in the very same unit, years, as the half-life.) Identify the worths recognized in the formula. Use the formula. Answer Tright here will certainly be 2.5 grams of caesium-137 in 60 years.
The provided text does not contain a mathematical question or a trouble that calls for a solution. It is an informative passage about exponential functions, their properties, and applications. The examples and also problems in the text are part of the explanation and also are not posed as inquiries for the reader to solve.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
In enhancement to direct, quadratic, rational, and also radical functions, tbelow are A feature of the form f(x) = bx, where b > 0 and b ≠ 1.

You are watching: Why do exponential functions grow faster than polynomials

")">exponential functions
. Exponential attributes have the create f(x) = bx, wright here b > 0 and b ≠ 1. Just as in any exponential expression, b is called the The expression that is being elevated to a power as soon as 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.

")">base
and also x is dubbed the When a number is expressed in the form ab, b is the exponent. The exponent shows just how many type of times the base is offered as a variable. Power and also exponent suppose the very same thing.

")">exponent
.

An example of an exponential attribute is the expansion of bacteria. Some bacteria double eextremely hour. If you begin via 1 bacterium and it doubles every hour, you will have 2x bacteria after x hrs. This can be created as f(x) = 2x.

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

After 1 hour f(1) = 21 = 2

In 2 hours f(2) = 22 = 4

In 3 hours f(3) = 23 = 8

and so on.

With the interpretation f(x) = bx and the restrictions that b > 0 and also that b ≠ 1, the doprimary of an exponential function is the collection of all actual numbers. The range is the collection of all positive actual numbers. The following graph mirrors f(x) = 2x.

Exponential Growth

As you deserve to view over, this exponential function has a graph that gets extremely close to the x-axis as the graph extends to the left (as x becomes more negative), however never before really touches the x-axis. Knowing the general shape of the graphs of exponential attributes is useful for graphing particular exponential equations or functions.

Making a table of worths is also beneficial, bereason you can use the table to area the curve of the graph even more accurately. One point 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 example,

.

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

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

x

 f(x) −2 −1 0 1 2

Choose numerous worths for x and put them as sepaprice rows in the x column.

Tip: It’s always excellent to encompass 0, positive values, and also negative values, if you have the right to.

x

 f(x) −2 −1 0 1 1 3 2 9

Evaluate the feature for each value of x, and compose the cause the f(x) column next to the x value you used. For example, when

x = −2, f(x) = 3-2 =

= , so  goes in the f(x) column next to −2 in the x column. f(1) = 31 = 3, so 3 goes in the f(x) column next to 1 in the x column.

Note that your table of values may be different from someone else’s, if you made a decision various numbers for x.

Look at the table of values. Think about what happens as the x worths increase—so perform the function values (f(x) or y)!

Now that you have actually a table of worths, you deserve to use these values to assist you attract both the shape and also area of the attribute. Connect the points as best you deserve to to make a smooth curve (not a series of right lines). This mirrors that every one of the points on the curve are component 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 creates an ordered pair that you can plot on a coordinate grid.

Plot the points.

Connect the points as ideal you can, using a smooth curve (not a series of directly lines). Use the form of an exponential graph to assist you: this graph gets exceptionally cshed to the

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

This is an instance of An exponential attribute of the form f(x) = bx, wright here b > 1, and also b ≠ 1. The function increases as x rises.

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

 Example Problem Graph f(x) = 4x. x f(x) −2 −1 0 1 1 4 2 16

Start with a table of values. You have the right to select different worths, yet as soon as aobtain, it’s useful to incorporate 0, some positive values, and some negative values.

Remember,

4-2 =

= .

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.

Notice that the larger base in this difficulty made the function value skyrocket. Even with x as little as 2, the function value is also huge for the axis scale you used before. You have the right to adjust the scale, yet then our other worths are exceptionally cshed together. You might additionally attempt various other points, such as once x =

. Due to the fact that you know the square root of 4, you deserve to discover that value in this case:
. The suggest
is the blue allude on this graph.

For various other bases, you could have to usage a calculator to aid you uncover the feature value.

Connect the points as ideal you can, utilizing a smooth curve (not a series of directly lines). Use the shape of an exponential graph to help you: this graph gets exceptionally cshed to the x-axis on the left, however never really touches the

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

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

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

Notice that a larger base renders the graph steeper. A bigger 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 attributes, b > 0, but b ≠ 1. In the examples over, b > 1. What happens when b is in 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 case,

, and
.

Use the table as ordered pairs and plot the points.

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

Notice that the form is similar to the shape as soon as b > 1, yet this time the graph gets closer to the x-axis once x > 0, rather than as soon as x An exponential feature of the develop f(x) = bx, wbelow 0 b . The feature decreases as x boosts.

")">exponential decay. Instead of the attribute values “growing” as x values boost, as they did prior to, the feature worths “decay” or decrease as x worths rise. They acquire 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. Again, 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 boost. The y values decrease.

Use the table pairs to plot points. You may want to incorporate new points, specifically once one of the points from the table, right here (−2, 16) won’t fit on your graph. Since you recognize the square root of 4, try x =. You can uncover that value in this case:

.

The suggest (, 8) has actually been consisted of in blue. You might feel it important to encompass extra points. You likewise may should use a calculator, depending on the base.

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

Which of the complying with is a graph for

?

A)

B)

C)

D)

A)

Incorrect. This graph is enhancing, because the f(x) or y values increase as the x values boost. (Compare the values for x = 1 and also x = 2.) This graph reflects exponential expansion, via a base higher than 1. The correct answer is Graph D.

B)

Incorrect. This graph is decreasing, yet all the function worths are negative. The selection for an exponential function is always positive values. The correct answer is Graph D.

C)

Incorrect. This graph is boosting, however all the attribute worths are negative. The correct graph must be decreasing through positive feature values. The correct answer is Graph D.

D)

Correct. All the attribute worths are positive, and also the graph is decreasing (showing exponential decay).

Applying Exponential Functions

Exponential attributes deserve to be offered in many conmessages, such as compound interest (money), population growth, and also radioenergetic decay. In many of these, yet, the attribute is not precisely of the form 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 A is the amount of money you would have, via interemainder, at the end of t years, making use of an yearly interemainder price of r (expressed as a decimal) and also m compounding periods per year. In this situation the base is the value represented by the expression 1 +
and also the exponent is mt—a product of 2 worths.

 Example Problem If you invest \$1,000 in an account paying 4% interest, compounded quarterly, how much money will certainly you have 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 recognize which of A, P, r, m, and also t is being asked for, then recognize values for the remaining variables. The primary is \$1,000. The rate is 4% = 0.04. The time in years is 3. Compounded quarterly suggests 4 times a year. To discover the amount A, use the formula. Answer You will certainly have actually \$1,126.83 after 3 years. Round the number to the nearemainder cent (hundredth). Notice that this implies the amount of interemainder earned after 3 years is \$126.83. (\$1,126.83, minus the principal, \$1,000).

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

Since the amount is halved each half-life, an exponential attribute can be used to explain the amount remaining over time. The formula

gives the remaining amount R from an initial amount A, wright here h is the half-life of the aspect and also t is the amount of time passed (utilizing the same time unit as the half-life).

 Example Problem Caesium-137 is a radioenergetic element offered 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, how much will certainly still be caesium-137 in 60 years? R: This is the remaining value, 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. (Note that this is in the very same unit, years, as the half-life.) Identify the worths recognized in the formula. Use the formula. Answer Tright here will certainly be 2.5 grams of caesium-137 in 60 years.
user
The provided text does not contain a mathematical question or a trouble that calls for a solution. It is an informative passage about exponential functions, their properties, and applications. The examples and also problems in the text are part of the explanation and also are not posed as inquiries for the reader to solve.

TYPE: NO_QUESTION