Algebra Nspirations
Algebra Nspirations
Algebra Nspirations
Algebra Nspirations
Algebra Nspirations
Algebra Nspirations
Exponents and Exponential Functions
The algebraic formula describing the shape of the
Saint Louis Gateway Arch.
The growth patterns of aesthetically pleasing
geometric shapes called fractals.
The rapid spread of disease among populations of humans,
animals, and plants.
The process called carbon dating archaeologists use to
estimate the age of artifacts.
And the disappearance of pattern of African elephants
due to loss of natural grazing land.
All use exponents in their algebraic expression or
functional equation.
Indeed, exponents and exponential functions are used
to describe these and other interesting objects and
processes in fields such as demography, biology,
chemistry, physics, economics, and finance.
We'll delve into some real world applications, but first
we'll look at exponents themselves by examining
notation, properties, and graphs.
Let's begin with some fun numbers.
A 31-year-old has lived over one billion seconds.
The number of oxygen atoms in an average sized
thimble is 1 octillion.
And a googol
not the internet search engine
is 1 followed by 100 zeros.
Now for some very small numbers.
A braid of 20 strands of fair hair is about
one millimeter thick.
A germ is typically one micrometer long.
And a water molecule is less than one
nanometer in diameter.
Multiplying an integer by 10 results in adding a zero.
So one billion is 10 multiplied by 10 nine times.
You may already know that mathematics has a shorthand
notation for this
10 to the ninth power
where the exponent denotes the number of factors.
Since 1 octillion has 27 zeroes, we
write 10 to the 27th.
And one googol is 10 the 100th power.
These are called powers of 10.
10 is the base, and the numbers at top
right are the exponents.
Here's the general notation for a power, any base, b.
Zero and negative exponents are easy to understand if we
look at the following patterns.
Notice from line to line, we're dividing by 10, thus
obtaining smaller and smaller powers of 10.
If you keep dividing by 10, we get 10 over 10, which is 1.
So 10 to the 0 is 1.
Dividing further, one over 10 is 10 to the negative 1.
1 over 10 times 10, or one over 10 squared, is 10 to the
negative 2.
One over 10 times 10 times 10, or one over 10 cubed, is 10 to
the negative 3.
And so on.
We can now use negative exponents to express a very
small number.
One millimeter is a thousandth of a meters.
1 over 1,000 is 10 to the negative 3.
Likewise, one micron or micrometer is a
millionth of a meter.
One over 10 to the sixth is 10 to the negative 6.
And one nanometer, used in nanotechnology, is one
billionth of a meter.
One over 10 to the ninth is 10 to the negative 9.
Scientific notation, a compact method of writing large and
small numbers favored by mathematicians, scientists,
and engineers, uses powers of 10.
For example, $10.71 trillion dollars, the total United
States federal debt as of February, 2009, can be written
in many equivalent ways.
The last line is in scientific notation, because the number
multiplying the power of 10 is between 1 and 10.
The wavelength of blue light, which is within the visible
light spectrum for humans, is about 475 nanometers.
These are all equivalent, but again, the last line is
scientific notation, because the number multiplying the
power of 10 is between 1 and 10.
In general, a number in scientific notation has the
form a times 10 to the n, where a real
number and n an integer.
If a is positive, a is greater than or equal to 1 but less
than 10, or greater than negative 10 and less than or
equal to negative 1 if a is negative.
Using an absolute value, we can combine both conditions
into one and write it like this.
This form allows for easy comparison of two numbers at
the same time, as the exponent gives
their order of magnitude.
Here's a fun problem.
I'm sure you'll quit your friends on it.
Take a sheet of paper this is a standard sheet of 20 pound,
recycled paper about 0.0038 inches thick.
Translating that into something you can visualize,
300 sheets of these stand about 1 inch tall.
Fold it once.
I have two layers.
Twice.
I have four layers.
Thrice.
I have eight.
Four times
16.
And after five times, I have 32 layers.
How high will this stack be after 50 folds?
Imagine a huge sheet of paper.
You'd have to cut and stack after a while, because folding
will become impossible.
Our doubling process here can be modeled with powers of 2.
n folds produces 2 the n layers.
So 50 folds would give a stack of 2 to the 50th layers, or
sheets of paper.
With 300 sheets equal 1 inch, 12 inches equal 1 foot, 3 feet
equals 1 yard, and 1,760 yards equal 1 mile, I'll let you do
the computation.
After just 50 folds, to do with that would be about 60
million miles high.
Yes, you heard it right.
That's more than 2/3 of the way from the Earth to the sun.
Mind boggling.
We've seen powers of 10 or 10 to the x, and powers of
2 or 2 to the x.
Let's turn now to the TI Inspire and use function
graphs to visualize the patterns of change of the
simplest exponential function of the form y equals b to the
x for a variety of base values b.
When the pattern increases, we have exponential growth.
When it decreases, it's exponential decay.
Finally, notice that we use b to the n for integer exponents
and b to the x for real exponents.
Turn on the TI Inspired.
Press the Home key for a new document.
If a document is open, you'll be prompted to save it.
After deciding, select 2 to create a graph
and geometry page.
To define f1 of x, enter the letter b for the base,
followed by the power or caret key.
And then x for the exponent.
Press Enter to graph.
No graph appears because we haven't defined values for b.
To do so, let's insert a slider.
Press Menu, and under Action, select Insert Slider.
Type b to replace the default variable, z1.
Press Escape or Tab and the graph will appear.
Notice the graph lies above the x-axis, meaning that the y
values of all points on the graph are positive.
Let's therefore modify the window settings.
Press Menu and under Window, select Window Setting.
Use the Tab key to scroll down from one line to the next.
Enter negative 8 and 8 for x.
Skip Auto, then negative 3 and 12 for y.
Tab down to OK.
Next, use the Nav Pad to move the pointer over the slide
bar, also called the thumb of the slider.
Press Control click to grab the thumb.
You should now see a closed hand.
Use the Nav Pad to slide the thumb along the slider track,
noting the changing values of b and how they affect the
shape of the exponential graph.
Since the tick marks on the track represent 2, 4, 6, and 8
along the continuous interval, 0 to 10, we couldn't see very
well what happened when b took on small values
between 0 and 1.
To better observe this, let's modify the slider setting.
Press Control, Menu to access the Slider Context Menu, and
select Settings.
Here again, use Tab to scroll down.
Skip b, then enter 0, negative 0.1, and 3.1.
For step size, select Enter Value by
pressing the down arrow.
Type 0.2 and then tab down to OK.
Again, slide the bar slowly along the
interval from 0 to 2.
Carefully observe what happens at b equals 0 and b equal 1.
And notice when the graph switches from a falling graph
to a rising graph
or from descending to ascending.
Stop at the graph corresponding to b equal 3.
One more property of exponential function graph
before summarizing our findings.
Press Menu, and under Points and Lines, select Point On.
Use the Nav Pad to drag the pointer to quadrant one.
Click to place one point at the top of the graph, another
in the middle, and a third lower down.
For each point, xy 3 to the x equals y.
Press Menu again, and this time, under Transformation,
select Reflection.
To reflect each point about the y-axis, drag the blinking
pencil to the y-axis first.
Click, then click on the point.
Repeat for each point.
Press Escape to exit this selection mode.
Notice the short decimal expansion of
numbers in this document.
You can do the same by pressing Home, then selecting
Document Settings.
Change your settings as you wish.
Next, press Control G to reveal the function entry
line, and enter the following for f2.
The left paren.
Control division for a fraction placeholder.
The fraction 1/3.
The right arrow.
The right paren.
They carrot key.
And finally x.
This is the exponential function in base 1/3.
Press enter to graph.
f2 of x passes through all three reflected points.
Notice how f1 of x equals 3 to the x, and f2 of x equals 1/3
to the x are reflections
or mirror images of each other
about the y-axis.
Can you figure out why?
Lastly, notice the intersection
point of the two graphs.
If y-coordinate is 1, and since it's on the y-axis, its
x-coordinate is 0.
In fact, all exponential functions of the form b to the
x pass through 0, 1.
Let's generalize what we've found in the form of
fundamental properties for all exponential functions of the
form b to the x.
One, exponential functions of the form b to the x are
defined for positive base values only.
We run into problems if b is negative.
We also exclude b equals 1, since 1 to the x
always equals 1.
Two, the domain is the set of all real numbers, since b to
the x is defined for all x.
Three, the range of the set of positive real numbers, since y
equals b to the x is always positive.
The graphs were all above the x-axis.
Four, all exponential functions b to the x have a
y-intercept of one.
Indeed, their graphs intersect the y-axis at the
point 0 comma 1.
Five, if b is greater than one, the functions increase.
And if b is strictly between 0 and 1, the functions decrease.
Six, if the bases are reciprocals of each other,
like 3 and 1/3, the graphs are symmetric about the y-axis.
The last result is explained by the following facts.
1 over b to the x equals b to the negative x, and x and
negative x are symmetric about the y-axis.
Reflect on that for a moment.
And if you need to, pause the video.
Before we investigative a finance application of
exponential functions, try this.