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1.02 Exponential functions and Euler's number

Lesson

In the investigation for this chapter we look at a special number, the mathematical constant, $e$e. We showed that $e$e is defined as $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^n$limn(1+1n)n and is the irrational number $2.7128\dots$2.7128. In our previous lesson we explored exponential functions of the form $y=a^x$y=ax, a very important member of this family of curves is $y=e^x$y=ex

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

 

Graphs of $y=2^x$y=2x$y=e^x$y=ex and $y=3^x$y=3x

 

Key features:

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

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

  • $y=0$y=0 is a horizontal asymptote for each graph. As $x\ \rightarrow-\infty$x , $y\ \rightarrow0^+$y 0+

  • Domain: $x$x is any real number

  • Range: $y>0$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 $e$e. Using a base of $e$e will become fundamental in exponential applications involving calculus. The function $f\left(x\right)=e^x$f(x)=ex has the remarkable property that $f'\left(x\right)=f\left(x\right)$f(x)=f(x) and we will see using a base of $e$e will often simplify required processes and calculations.

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

Worked example

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

Think: For the function $y=A\times e^{\left(bx-h\right)}+k$y=A×e(bxh)+k, what impact does each parameter have? Which parameters have been altered?

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

b) Sketch the function.

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

Do:

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

 

Practice questions

Question 1

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

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

    For $x>\editable{}$x>, the graph of $y=e^x$y=ex will lie above the graph of $y=\left(\editable{}\right)^x$y=()x and below the graph of $y=\left(\editable{}\right)^x$y=()x.

    For $x<\editable{}$x<, the graph of $y=e^x$y=ex will lie above the graph of $y=\left(\editable{}\right)^x$y=()x and below the graph of $y=\left(\editable{}\right)^x$y=()x.

Question 2

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

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

  2. The function is then translated $2$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 $y$y-intercept of the new function?

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

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    A

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    B

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    C

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    D

Outcomes

3.2.1.3

define the exponential function 𝑒^x

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