Graphing exponentials of the form y=b^x with 0<b<1

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Xiaodan Leng 5 years, 9 months ago

Created this as a copy of Graphing exponentials of the form y=b^x with 0<b<1.

Name	Status	Author	Last Modified
Graphing exponentials of the form y=b^x with b>1	Ready to use	Ben Brawn	07/03/2025 05:47
Graphing exponentials of the form y=b^x with 0<b<1	Ready to use	Ben Brawn	07/03/2025 05:47
Graphing exponentials with vertical transformations and base>1	Ready to use	Ben Brawn	07/03/2025 05:47
Graphing exponentials with horizontal and vertical transformations and base>1	Ready to use	Ben Brawn	07/03/2025 05:47
Graphing logarithms of the form y=log_{b}(x) with b>1	Ready to use	Ben Brawn	03/03/2025 00:25
Graphing logarithms with vertical transformations and base>1	Ready to use	Ben Brawn	03/03/2025 00:25
Graphing logarithms with horizontal and vertical transformations and base>1	Ready to use	Ben Brawn	03/03/2025 00:25
Jamie's copy of Graphing logarithms of the form y=log_{b}(x) with b>1	draft	Jamie Wood	19/09/2017 12:36
Jamie's copy of Graphing logarithms with vertical transformations and base>1	draft	Jamie Wood	19/09/2017 12:38
Jamie's copy of Graphing exponentials with horizontal and vertical transformations and base>1	draft	Jamie Wood	19/09/2017 13:33
Jamie's copy of Graphing exponentials with vertical transformations and base>1	draft	Jamie Wood	19/09/2017 13:29
Jamie's copy of Graphing logarithms with horizontal and vertical transformations and base>1	draft	Jamie Wood	19/09/2017 12:43
Jamie's copy of Graphing exponentials of the form y=b^x with b>1	draft	Jamie Wood	19/09/2017 13:07
Maria's copy of Graphing exponentials of the form y=b^x with b>1	draft	Maria Aneiros	25/05/2019 08:02
Exam04 - Exponentials	draft	Michael Proudman	07/12/2021 09:54
nuExam04 - Exponentials	draft	Michael Proudman	07/12/2021 09:54
Graphing exponentials with vertical transformations and base>1	draft	Xiaodan Leng	11/07/2019 01:13
Graphing exponentials with vertical transformations and base>1	draft	Xiaodan Leng	11/07/2019 01:26
Graphing exponentials with horizontal and vertical transformations and base>1	draft	Xiaodan Leng	11/07/2019 01:27
Graphing exponentials of the form y=b^x with 0<b<1	draft	Xiaodan Leng	11/07/2019 01:31
Graphing exponentials of the form y=b^x with b>1	draft	Xiaodan Leng	11/07/2019 01:32
Graphing logarithms with horizontal and vertical transformations and base>1	draft	Xiaodan Leng	11/07/2019 02:17
Graphing logarithms of the form y=log_{b}(x) with b>1	draft	Xiaodan Leng	11/07/2019 02:18
Graphing logarithms with vertical transformations and base>1	draft	Xiaodan Leng	11/07/2019 02:18

There are 20 other versions that do you not have access to.

Question statement

The following questions will gauge your understanding of exponentials and how to graph them.

The exponential you will be working with for this question is

$y=\left(\simplify[fractionNumbers]{{b}}\right)^x.$

Give any introductory information the student needs.

			Name	Type	Generated Value

rational

3/10

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1/10,1/5,3/10,2/5,1/2,3/5,7/10,4/5,9/10,1/3,1/4,1/6,1/7,1/8)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `rational`

3/10

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
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Part b) Gap-fill
1. Gap Number entry
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Part c) Gap-fill
1. Gap Choose one from a list
  Add
Part d) Choose one from a list
Part e) Gap-fill
1. Gap Number entry
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Part f) Choose one from a list

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

The $y$-intercept of $y=\left(\simplify[fractionNumbers]{{b}}\right)^x$ is the point $\large($Gap 0., Gap 1.$\large)$.

Advice

Note: if we so desire we can rewrite this exponential with a negative index since $\left(\simplify[fractionNumbers]{{b}}\right)^x=\left(\simplify[fractionNumbers]{{1/b}}\right)^{-x}$.

a) To find the $y$-intercept, substitute $x=0$ into the equation: $y=\left(\simplify[fractionNumbers]{{b}}\right)^0=1$. Therefore, the $y$-intercept is the point $(0,1)$.

b) Substitute $x=1$ into the equation: $y=\left(\simplify[fractionNumbers]{{b}}\right)^1=\simplify[fractionNumbers]{{b}}$. Therefore, another easily found point is $\left(0,\simplify[fractionNumbers]{{b}}\right)$.

c) Let's investigate what happens to the value of $y$ when we add 1 to the value of $x$:

\[\left(\simplify[fractionNumbers]{{b}}\right)^{x+1}=\left(\simplify[fractionNumbers]{{b}}\right)^x\left(\simplify[fractionNumbers]{{b}}\right)^1=\left(\simplify[fractionNumbers]{{b}}\right)^x\left(\simplify[fractionNumbers]{{b}}\right)\] That is, the old $y$ value is multiplied by $\simplify[fractionNumbers]{{b}}$ (which actually decreases the $y$ value since $\simplify[fractionNumbers]{{b}}<1$).

d) Since $y=\left(\simplify[fractionNumbers]{{b}}\right)^x$ is an exponential and as $x$ increases $y$ decreases to 0, we call this exponential decay.

e) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\left(\simplify[fractionNumbers]{{b}}\right)^x$ the larger $x$ gets, the closer $y$ gets to $0$. In other words as $x$ approaches infinity, $y$ approaches $0$. This means that the asymptote for $y=\left(\simplify[fractionNumbers]{{b}}\right)^x$ is the line $y=0$ (the $x$-axis).

f) Given all the information above, it should be clear that the graph should look like

{graph1(1)}

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
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Gap-fill		Hasn't run yet
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Graphing exponentials of the form y=b^x with 0

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `rational`

Parts

Gap-fill

Loading...

Error

Graphing exponentials of the form y=b^x with 0

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: rational

Parts

Gap-fill

Generated value: `rational`