Professor Dave again, let’s learn about exponential functions.
We’ve learned about a variety of functions, including ones with exponents in them, like
X squared.
But what if these two items swapped positions, and we instead had two to the X power?
This would be called an exponential function, because a variable is acting as an exponent.
To see what these do, let’s plot a few points.
When X is zero, Y is one, because anything to the zero power is equal to one, which we
learned when we went over the rules of exponents.
When X is one, Y is two, because two to the one is two.
When X is two, Y is four, because two squared is four.
When X is three, Y is eight, and so forth.
As we can see, the function grows very rapidly as X increases.
This number that is raised to the exponent is called the base, and changing the base
will produce different exponential functions.
We could have ten to the X, which will increase by an order of magnitude every time X increases
by one.
The exponent can also be an expression, like with three to the quantity X minus two.
The only parameter is that the base must be positive and not equal to one.
Exponential functions are valuable, because they can model a number of real-world situations
where rapid increase or decrease occurs.
Population growth can often be modeled by an exponential function, especially for something
like bacterial cultures, or the spread of diseases.
They are easy to evaluate on your calculator.
For something like 1.5 times four to the X, if we want to evaluate this for X equals three,
we just cube four to get sixty-four, and then multiply by 1.5 to get ninety-six.
Graphing can be a little trickier, but as long as we know what the basics look like,
we can apply transformations just like we did for parabolas.
Once again, two to the X looks like this.
It has a horizontal asymptote in the X axis, and the function will always be positive,
because there is no exponent that two can be raised to in order to get a negative number.
Negative exponents, as we recall, still produce positive values, as two to the negative one
is just one half.
Two to the negative two is one over two squared, or one fourth.
And so on, getting closer to zero as the exponent increases in the negative direction.
In the positive direction, as the exponent increases, the function approaches positive
infinity.
Now let’s look at one half to the X.
Now, as the exponent increases, the function approaches zero, so there is again a horizontal
asymptote in the X axis.
As the exponent becomes increasingly negative, the function approaches positive infinity.
This is actually the same as two to the negative X, because to get rid of the negative sign
on the exponent, we invert the base, which would give us one half to the X.
This means that for exponential functions, if the base is less than one and the exponent
is positive, or if the base is greater than one and the exponent is negative, the function
dips down to the right like this.
If the base is greater than one and the exponent is positive, or if the base is less than one
and the exponent is negative, the function will rise up to the right, like the previous
example.
But in both cases, the domain is all real numbers, and the range is all real numbers
greater than zero.
As we said, we can graph alterations of this basic form by applying transformations.
Adding or subtracting a number here will generate a vertical shift.
Adding or subtracting a number from the variable term right here, like two to the X plus one,
will produce a horizontal shift.
If we put a negative sign here, we get a reflection across the X axis, since all the positive
values become negative values.
If instead we put the negative sign here, before the variable, we get a reflection across
the Y axis, just as we saw earlier with two to the negative X.
And lastly, a coefficient here or here will produce vertical stretching and horizontal
stretching, respectively. Let's check comprehension.