Graphing exponentials of the form y=b^x with 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 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

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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

$y=\var{b}^x.$

Give any introductory information the student needs.

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integer

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xxxxxxxxxx
 
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Generated value: `integer`

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Parts

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

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

The $y$-intercept of $y=\var{b}^x$ is the point $\large($Gap 0., Gap 1.$\large)$.

Advice

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

b) Substitute $x=1$ into the equation: $y=\var{b}^1=\var{b}$. Therefore, another easily found point is $(1,\var{b})$.

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

\[\var{b}^{x+1}=\var{b}^x\var{b}^1=\var{b}^x\var{b}\] That is, the old $y$ value is multiplied by $\var{b}$, so we can say that $y$ is increased by a factor of $\var{b}$.

d) Since $y=\var{b}^x$ is an exponential and as $x$ increases $y$ increases without bound, we call this exponential growth.

e) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\var{b}^x$ the smaller $x$ gets, the closer $y$ gets to $0$. In other words as $x$ approaches negative infinity, $y$ approaches $0$. This means that the asymptote for $y=\var{b}^x$ 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)}

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

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Taxonomy: mathcentre

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Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

Loading...

Error

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

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: integer

Parts

Gap-fill

Generated value: `integer`