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4.05 The natural exponent

Lesson

We will now look at a special number, the mathematical constant, $$e. $$e is defined as $$limn(1+1n)n and is the irrational number $$2.7128. In a previous lesson we explored exponential functions of the form $$y=ax, a very important member of this family of curves is $$y=ex

As a member of this family of curves we can see it has the same properties and sits between the functions $$y=2x and $$y=3x.

 

Graphs of $$y=2x$$y=ex and $$y=3x

 

Key features:

  • Exponential growth: As the $$x-values increase, the $$y-values increase at an increasing rate.

  • The $$y-intercept is $$(0,1), since when $$x=0$$y=a0=1, for any positive value $$a.

  • $$y=0 is a horizontal asymptote for each graph. As $$x , $$y 0+

  • Domain: $$x is any real number

  • Range: $$y>0

The three functions above can be viewed as horizontal dilations of each other. Using transformations we can in fact write any exponential function with a base of $$e. Using a base of $$e will become fundamental in exponential applications involving calculus. The function $$f(x)=ex has the remarkable property that $$f(x)=f(x) and we will see using a base of $$e will often simplify required processes and calculations.

Just as with functions of the form $$y=ax we can perform transformations on the graph $$y=ex.

Worked example

a) For the function $$y=ex, describe the transformations required to obtain the graph of $$y=2×ex+3.

Think: For the function $$y=A×e(bxh)+k, what impact does each parameter have? Which parameters have been altered?

Do: We have $$A=2 and $$k=3, hence, the function has been vertically dilated by a factor of $$2 and a vertical translation by a $$3 units upwards.

b) Sketch the function.

Think: From the basic graph of $$y=ex a vertical dilation by a factor of $$2 will stretch each point away from the $$x-axis by a factor of $$2. Hence, the point $$(0,1) becomes $$(0,2), $$(1,e) becomes $$(1,2e) and so forth. Then we can shift the graph $$3 units vertically, each point moves up $$3 units and the horizontal asymptote becomes $$y=3.

Do:

Step 1. Dilate the graph by a factor of $$2 from the $$x-axis. Step 2. Translate the graph upwards $$3 units, this includes the horizontal asymptote.

 

Practice questions

Question 1

The functions $$y=2x and $$y=3x have been graphed on the same coordinate axes.

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  1. Using $$e=2.718 and by considering the graph of $$y=ex, complete the statement below:

    For $$x>, the graph of $$y=ex will lie above the graph of $$y=()x and below the graph of $$y=()x.

    For $$x<, the graph of $$y=ex will lie above the graph of $$y=()x and below the graph of $$y=()x.

Question 2

Using a graphing calculator, graph the curves of $$y=ex, $$y=ex+2, and $$y=ex3 on the same screen.

  1. Select the correct statement from the following:

    The three graphs are the same.

    A

    The three graphs have the same shape.

    B

    The three graphs have the same $$y-intercepts.

    C

    The three graphs have the same $$x-intercepts.

    D

Question 3

Beginning with the equation $$y=ex, we want to find the new function that results from the following transformations.

  1. Starting from $$y=ex, the function is first dilated by a factor of $$3 vertically. What is the equation of the new function?

  2. The function is then translated $$2 units upwards. What is the equation of the new function?

  3. What is the equation of the horizontal asymptote of the new function?

  4. What is the value of the $$y-intercept of the new function?

  5. Using the previous parts, choose the correct graph of the transformed function.

    Loading Graph...

    A

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    B

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    C

    Loading Graph...

    D

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