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 $$y=3x |
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Key features:
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The three functions above can be viewed as horizontal dilations of each other. Using transformations we can in fact write any exponential function with a base of $$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.
a) For the function $$y=ex, describe the transformations required to obtain the graph of $$y=2×ex+3.
Think: For the function $$y=A×e(bx−h)+k, what impact does each parameter have? Which parameters have been altered?
Do: We have $$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 $$3 units, this includes the horizontal asymptote. |
The functions $$y=2x and $$y=3x have been graphed on the same coordinate axes.
Using $$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 $$y=()x.
Using a graphing calculator, graph the curves of $$y=ex, $$y=ex+2, and $$y=ex−3 on the same screen.
Select the correct statement from the following:
The three graphs are the same.
The three graphs have the same shape.
The three graphs have the same $$y-intercepts.
The three graphs have the same $$x-intercepts.
Beginning with the equation $$y=ex, we want to find the new function that results from the following transformations.
Starting from $$y=ex, the function is first dilated by a factor of $$3 vertically. What is the equation of the new function?
The function is then translated $$2 units upwards. What is the equation of the new function?
What is the equation of the horizontal asymptote of the new function?
What is the value of the $$y-intercept of the new function?
Using the previous parts, choose the correct graph of the transformed function.