Eeeeeeeeeeeeeee!!!!! It’s ______________________________!
Date _______________________
Period ____________
The Natural Base, e
H – Algebra 2
Let’s examine two of our more recent concepts together; let’s think about graphs of exponential functions and
sums of infinite geometric series together.
EXPONENTIAL FUNCTIONS
We saw very recently how to graph an exponential function, y = bx, and how to discuss its end behavior.
If b < 1, we will see exponential decay.
1
y 
2
If b > 1, we will see exponential growth.
x
y  2x
as x  ∞, y  __________
as x  ∞, y  __________
INFINITE GEOMETRIC SERIES
We reasoned recently that if we add up an infinite quantity of numbers, the answer may approach infinity.
EX]
1 + 3 + 5 + 7 + 9 + ··· → ∞
But then we wondered what would happen if the terms we added together got closer to zero, instead of further.
EX]
1 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + ··· = 2
It turns out that doing something an infinite number of times does NOT always give us infinity as a result.
So, that brings us to this question: What happens if we take a number that is ever so slightly larger than one (the
difference is so small we can’t even measure it) and multiply it by itself an infinite number of times?
n

1
 1

What is 1   as n gets closer and closer to ∞? What is 1   ?
 
 n
To answer this question, we will use our calculators. We won’t get the exact answer, but we will get very close.
Option 1:
Choose an arbitrarily large number for n. How about a thousand? Probably too small. A million? Still too
small. A billion? Closer, but I’m thinking more like _____________________. Now that is a pretty darn big
number, until you compare it to infinity (or our national debt, but that is another discussion altogether.) Plug
this value in for n into your calculator.
Our approximate answer: ___________________________
Option 2:
Plug the expression into “y =” on your graphing calculator. “y = (1 + 1/x)^x”. Hit the “window” button, and
set x-min to 0, x-max to 10, y-min to 0, y-max to 10. Press “graph”. The graph appears to be near horizontal
on the right side. Press “Trace”. Use the arrows to move the cursor left and right; does the y-value stay the
same? Or does it change?
As we move to the right, the y-value should get higher and higher. Unfortunately, we can’t move it over far
enough to the right.
Press “window” again. Set x-min to 9,000,000 and the x-max to 10,000,000. Hit “graph”, and again “trace”
your cursor along the curve. Unfortunately, when we do this, our y-value does not steadily increase. Instead, it
kind of goes up and down very small amounts. THIS IS A LIMITATION OF THE CALCULATOR!!! It does
not quite have enough memory to perform the calculations with numbers this big while graphing!!
Nonetheless, our approximate answer: ___________________________
It appears that Option 1 gave us a better result. Run through Option 1 again, so we all have the answer on-screen.
Now find the button for “e” on your calculator (“2nd – division” for plain e, or “2nd – LN” to open up an exponent).
Hit enter, and display the value of e. What do you notice?
It turns out this e thing is a CONSTANT; its value does not change. e is also irrational; it is a non-repeating
decimal that goes on forever. Can you think of any other irrational constants symbolized by a single character?
e is very important in mathematics, and its name is the Natural Base. It got this name because, you guessed it, it
shows up in many natural phenomena involving exponential growth and decay. Since it is the Natural Base, and
we are currently graphing exponential functions, we will need to graph y = ex and its transformations.
Let’s look at some powers of e, and their decimal approximations.
x
y = e (exact)
y = ex (approx.)
x
-1
e-1
0.36788
0
1
1
1
e
2.7183
2
e2
7.3891
3
e3
20.086
4
e4
54.598
What we should notice is that the only EXACT value that is easily graphed is 1, as in (0, 1). The others are too
difficult to graph by hand, purely from memory. Therefore, will need to know exactly for what we are responsible
when graphing transformations of y = ex.
We will need to know/graph the following for transformations of y = ex
1. The asymptote. (y = vertical shift. This is the same as before.)
2. The domain: (-∞, ∞) This is the same as before.
3. The range:
a. (asymptote-value, ∞) for y = (+)ex
b. (-∞, asymptote-value) for y = (-)ex
c. These are the same as before
4. Decay vs. Growth
a. Growth: y = ex
b. Decay: y = e-x = (1/e)x
c. These are the same as before, when we recall that a negative exponent reciprocates the base.
5. Where the parent-point (0, 1) is after all transformations. This will behave the same way as before.
Examples:
1. y = -3ex-2 – 4
Parent Function: __________ Asymptote: _________
Transformation: ____________________________
_____________________________
____________________________
Domain: ____________
Range: _____________
Growth/Decay: _____________________________
End Behavior:
as x  , y  ___________
as x  , y  ____________
2. y = 4e-(x+3) +5
Parent Function: _________ Asymptote: __________
Transformation: ____________________________
_____________________________
____________________________
Domain: ____________
Range: _____________
Growth/Decay: _____________________________
End Behavior:
as x  , y  ___________
as x  , y  ____________
3. y = 2ex+3 + 1
Parent Function: __________ Asymptote: _________
Transformation: ____________________________
_____________________________
____________________________
Domain: ____________
Range: _____________
Growth/Decay: _____________________________
End Behavior:
4. y = -7e-(x-2) +5
as x  , y  ___________
as x  , y  ____________
Parent Function: _________ Asymptote: __________
Transformation: ____________________________
_____________________________
____________________________
Domain: ____________
Range: _____________
Growth/Decay: _____________________________
End Behavior:
as x  , y  ___________
as x  , y  ____________
5. y = 3e-(x+4) – 6
Parent Function: __________ Asymptote: _________
Transformation: ____________________________
_____________________________
____________________________
Domain: ____________
Range: _____________
Growth/Decay: _____________________________
End Behavior:
6. y = -2ex-3 – 4
as x  , y  ___________
as x  , y  ____________
Parent Function: _________ Asymptote: __________
Transformation: ____________________________
_____________________________
____________________________
Domain: ____________
Range: _____________
Growth/Decay: _____________________________
End Behavior:
as x  , y  ___________
as x  , y  ____________