WEBVTT
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Let's look at a new type of function.
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An exponential function is a function that
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can be written in this form: f of x equals a
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to the x where a is a positive real number,
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which means it's greater than 0, but it can't be 1.
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The domain of this function will be the set of all real numbers.
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What makes this different from what you're used to?
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Mostly it's the placement of the variable.
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Notice how the variable is in the exponent
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now, and you have a real number for the base.
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This is different from the algebraic
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functions that we've studied so far.
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We have a couple of restrictions on what that base can be.
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Why are they there? Well, we don't want a to equal 1 because
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that would be the constant function 1 to the
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x, which is kind of boring, and it's also linear.
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Okay we've already studied linear functions.
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Why do we want a to be greater than 0?
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Well, what if it was negative 4? Can you raise negative 4 to the 1/2 power?
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Not if you want a real number answer, so in
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order to have the domain be the set of all
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real numbers, we want to restrict our base, a, to be something bigger than 0 and to not
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be linear, we don't want a to be 1.
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Let's look at a specific exponential function.
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Here we have the exponential function: f of x equals 2 to the x.
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To get used to the idea of an exponential
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function and how they work, let's find some specific function values.
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First, we want to find f of 0. So, we want to find the value of this
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function when x is 0. We need to find 2 to the 0 power which is 1,
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so f of 0 is 1. The next one: when x is 1,
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what's the value of the function?
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So, we need to find 2 to the first power which is 2.
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f of 3: we want to find 2 to the third power.
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So, you can see as we go along that our
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base, 2, doesn't change, but our exponent
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changes according to what x is.
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Now let's try something with a negative number.
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Here it asks us to find the value of 2 to the x when x is negative 1.
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That’s 2 to the negative 1 power, and you'll have to
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remember some rules of exponents.
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We can make the exponent positive by writing
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this as 1 over 2 to the first power or 1/2.
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If x is negative 2, we have to find 2 to the negative 2.
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That's 1 over 2 squared or 1/4.
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Now what about these two? Well, we start out the same way.
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We want to find the value of the function when x is the square root of 3.
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So, we need 2 to the square root of 3 power.
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We can't do that in our head, but we can use
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a calculator to approximate the value.
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What you want to look for on your calculator
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is a button that will have an x to the y or perhaps the carot.
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Those are the two most common ways to find this on your calculator.
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We're going to start by entering the 2 and
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then raise it to the square root of 3 power.
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We'll round off to three decimal places, so
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this is approximately 3.321, and actually we
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want to round that off at the end, so let's try 3.322.
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We have one more to try. f of negative 1.2 is 2 to the negative 1.2 power.
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Be careful to make sure you don't leave off
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the negative sign in the exponent,
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and then round off to three decimal places,
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so this will be approximately 0.435.
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We have all these function values, so
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let's try to plot the graph, plot some
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points and draw a graph for this function.
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We're going to use the ones that are easiest
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to plot, so we know from this one that when
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x is 0, the value of the function is 1.
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We know here that when x is 1, the y value, the function value is 2.
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When x is 3, the function value is 8.
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If you think about it for a moment, you know
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that the bigger you take x to be, then
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that's a bigger power on 2, so this is going
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to continue to get bigger and bigger without bound.
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So, we can finish this side of the graph like this.
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What about on the other side? Well, we have a couple of points
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that will help us. When x is negative 1, the function value is 1/2.
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When x is negative 2, the function value is 1/4, so
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it's getting smaller and smaller, closer and closer to 0.
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If you keep going, will it ever cross the x-axis?
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No, because it'll get smaller and smaller,
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but it's always going to end up being a positive number.
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So, we can draw the rest of the graph in
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this manner, showing that it gets closer and
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closer to the x-axis without ever touching or crossing it.
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In this case, we call the x-axis a
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horizontal asymptote for the graph.
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We'll return to this type of exponential function in a moment.
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Here's one more exponential function to explore.
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Notice, it looks a little different.
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Here the base is a fraction between 0 and 1.
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Let's start out the same way and find some specific function values.
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f of 0 is the value of the function when x is 0.
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This gives me 1/2 to the 0 power, which is 1.
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f of 1 is going to be 1/2 to the first power, that's 1/2.
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f of 3: find the value of the function when
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x is 3, that's 1/2 to the third power or 1/8.
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With the negative values of x, you're just
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going to have to remember those rules of exponents again.
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You want to find 1/2 to the negative 1, this is the
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same as 2 to the first or 2. For f of negative 2, we need to find 1/2 to the
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negative 2 power, which is the same as 2 to the second power or 4.
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Let's go ahead and plot the points and see what this graph looks like.
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Our first point: when x is 0, y is 1.
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This one gives me when x is 1,the y value is 1/2.
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When x is 3, the y value is 1/8.
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So, again we see that behavior we saw before, just on the other side.
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The larger we choose x to be, the closer to 0 the function value will get.
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But, just as before, it will not ever equal
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zero or become negative, so we have another horizontal asymptote.
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Now let's go the other way. When x is negative 1, the value of the function is 2.
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When x is negative 2, the value of the function is 4.
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You can imagine what's going to happen
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if you keep picking negative 3, negative 4, negative 5 as your x
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value, you're going to get a larger and larger value of y.
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So you can see that depending on what the
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value of your base is, you can have a slightly different looking graph.
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Let's explore this in general a little bit more.
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Let's take a closer look at the graphs of exponential functions.
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Depending on the value of the base, a, there
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are two different cases that will come up.
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If the base, a, is greater than 1, then the graph will have this shape.
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Notice that it goes through the point (0,1),
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the point (1,a), and then the point (-1,1/a).
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The second case is when a is between 0 and
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1, and the graph is a little bit different. It'll still have the point (0,1).
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It'll go through the point (1,a), but a is a
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fraction between 0 and 1 or a decimal, and
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that's going to be something very small, very close to the x-axis.
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It's also going to have the point (-1,1/a),
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but again a is a very small number, and
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its reciprocal will be a very large number.
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So, the graph of f of x equals a to the x
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has one of these two shapes, depending on a.
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Let's write down some of these characteristics.
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First of all, we want to know the domain of the function.
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Well, that doesn't depend on a: the domain
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is all real numbers for each of these. What about the range?
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The range is the set of allowed y values for
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the function, and in both of these cases, y can be anything greater than 0.
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So, I'll write that in interval notation: y can be anything greater than 0.
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Both of the graphs have a y-intercept at 1.
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In fact, neither of them has an
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x-intercept because neither of them ever touch the x-axis or cross it.
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Because neither of them touch or cross the
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x-axis and we've already discussed how the
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graph gets closer and closer to the x-axis
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without touching or crossing and we call
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that an asymptote, both of them have a horizontal asymptote.
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The asymptote is the x-axis, the
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equation of the x-axis is y equals 0.
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One difference that you can see is that for
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a, greater than 1, you get an increasing function: it rises as you go
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from left to right. In the case of a being between 0 and 1, you
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have a decreasing function that falls from left to right.
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The last characteristic is these are
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both, in either case, a one-to-one function,
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and that's an important observation to make as well.
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Now let's look at using transformations to graph exponential functions.
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Now that we've seen how to graph basic
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exponential functions, let's use this to
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graph more complicated exponential
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functions using transformations.
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So, our instructions here are the sketch the
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graph using transformations, to state the
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domain, the range, and the asymptote.
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Alright, so the first thing we want to do is
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find the basic exponential involved, and that's 3 to the x.
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Let's graph that first. The graph of an exponential function has to
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contain the point (0,1). When x is 0, 3 to the 0 is 1.
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It's also going to have to contain the point
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(1,3) because, when x is 1, 3 to the 1 is 3.
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Keeping in mind the basic shape, we get a graph that looks like this.
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So, as we said earlier, every exponential
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function contains the point (0,1) and the
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point 1 and the base, which happens to be 3 in this case.
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Now we're going to look at our transformation: we're subtracting 2
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from that exponential function. What does that do to the graph?
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Right, it's going to move the whole thing down 2 units.
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Okay, but we need to be careful because, as
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soon as we start moving this down, the
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asymptote has to be moved down first.
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Okay, so right now it's at y equals 0.
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I'm going to move that down 2 units,
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that's going to give me a new asymptote that
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I'm going to represent with a dotted line.
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Then I can move each point on the graph down 2 units.
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So, that's going to put this one at (0,-1),
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and this one moved down 2, we'll put it at (1,1).
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I haven't changed anything about the shape
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of the graph, I just slid it down.
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Now we've already basically said what the
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asymptote is because we moved it down 2 units: it's y equals negative 2,
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and it is a horizontal asymptote.
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The domain: it's still all real numbers
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because this is an exponential function,
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we haven't changed that. What about its range?
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Well, that will be different because when we
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slid it down 2 units, we've allowed for some
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smaller y values than we had before, they can be anything bigger than negative 2.
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So, write that in interval notation. Let's try one more.
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The next example shows us some other
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transformations and how they work with exponential functions.
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Again, the first thing I want to do is
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decide which basic exponential function is
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involved, and we're going to start with 2 to the x.
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I'll sketch the graph of this one. It has to have the point (0,1).
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It has to have the point 1 and the base,
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that's 2, so we're going to have the point (1,2).
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Keeping in mind that the basic shape for an
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exponential function's graph: when the base
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is bigger than 1, we can draw this.
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Now we're ready to think about those transformations.
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The first thing I see is the negative sign
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in front of the 2. What does that do to my graph?
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Right, it's a reflection about the x-axis.
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What about the plus 1 that's in the exponent?
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Right, that is a horizontal shift. Which way, left or right?
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Okay, so it's going to be shifted 1 unit to the left.
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Let's just make a note of that.
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We're going to go 1 to the left and reflect across the x-axis.
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If you're very careful, I think you can
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get this graph in one step. You know exactly what it should look like.
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We want to move this point to the left one
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unit, but then we're going to flip it over. So, where will It end up?
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Right, exactly on the other side of the x-axis.
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Okay, let's try this one: we're going to go
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to the left 1 and then reflect this across
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the x-axis, so where does this point end up? Right, on the y-axis at -2.
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The shape of the graph doesn't change, we've
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just flipped it over and slid it to the left.
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Okay, so now we need to think about the
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domain, the range, and the equation of the asymptote.
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Let's do the asymptote first. Notice that it gets closer and closer to the
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x-axis without touching it. It's below, but that's okay, it's still an
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asymptote at the x-axis, and the equation of the x-axis is y equals 0.
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The domain: it's still all real numbers. What about the range?
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Since it's the values y is allowed to have,
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it can be anything that is smaller than 0.
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So, in interval notation, it looks like this.
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We can also solve equations involving exponential functions.
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Let's look at a couple. Notice that it's nothing like an equation
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you've solved before because the variable
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that you're looking for is in the exponent.
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So, the idea to solving these is to rewrite
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so that both sides have the same base.
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If you look at the left hand side, this is 5 to the 1 minus 2 x.
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So, now look at the right hand side. Is 125 a power of 5?
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Yes, it is, it's 5 to the third, so we can
000:19:03.842 --> 000:19:07.841
rewrite this as 5 to the 1 minus 2 x equals
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5 to the third. Exponential functions are
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one-to-one functions so that's important to
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know because it says that if 5 to this power
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equals 5 to the third, the only way that can
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happen is if the powers are in fact equal.
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Now we can solve this linear equation.
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Subtract 1 from both sides, divide by negative 2, we find that x is negative 1.
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I won't worry about checking this one
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since the domain of an exponential function is all real numbers.
000:19:46.100 --> 000:19:50.133
As long as I don't make a careless mistake, my answer should be okay.
000:19:50.133 --> 000:19:52.884
Let's try the second example the same way.
000:19:52.884 --> 000:19:54.883
I would like for both sides to be written as
000:19:54.883 --> 000:19:57.134
powers of the same number, and right away I
000:19:57.134 --> 000:20:05.633
can see that 4 is 2 squared and 8 is 2 to the third, so let's use that.
000:20:05.633 --> 000:20:11.500
So 4 becomes 2 squared and then that's raised to the x squared power.
000:20:11.500 --> 000:20:17.133
Eight is 2 to the third which is raised to the x power.
000:20:17.133 --> 000:20:19.884
Remember my rules of exponents: when I raise
000:20:19.884 --> 000:20:23.850
a power to a power, I need to multiply those powers.
000:20:23.850 --> 000:20:29.000
So, this'll be 2 to the 2 x squared equals 2
000:20:29.000 --> 000:20:34.884
to the 3x, same property of exponents on the other side.
000:20:34.884 --> 000:20:36.885
Now, because exponential functions are one
000:20:36.885 --> 000:20:39.134
to one, the only way that 2 to this power
000:20:39.134 --> 000:20:47.000
equals 2 to this other power is if those powers are exactly the same.
000:20:47.000 --> 000:20:51.200
So, I have 2 x squared has to equal 3 x for
000:20:51.200 --> 000:20:53.202
this to be true, and I want to solve this.
000:20:53.202 --> 000:20:56.952
What kind of equation is it? It's quadratic, and we know
000:20:56.952 --> 000:20:59.458
how to solve those. We're going to put it in standard form which
000:20:59.458 --> 000:21:07.216
means set it equal to 0. How can I solve that?
000:21:07.216 --> 000:21:09.200
We're going to try to factor it, and in this
000:21:09.200 --> 000:21:18.200
case, I see that there's a common factor of x, so let's take it out.
000:21:18.200 --> 000:21:23.470
So, I get two solutions: x equals 0 and
000:21:23.470 --> 000:21:28.471
then, if 2 x minus 3 equals 0, by adding 3
000:21:28.471 --> 000:21:34.721
to both sides and dividing by 2, I get that x would have to be 3/2.
000:21:34.721 --> 000:21:38.500
Now let's look at one very special exponential function.
000:21:40.000 --> 000:21:44.503
Let's look at this expression: 1 plus 1 over n
000:21:44.503 --> 000:21:47.754
raised to the n power. What I'd like to know is what happens to
000:21:47.754 --> 000:21:50.504
this expression as n gets larger and larger.
000:21:50.504 --> 000:21:52.512
So, let's take a couple of values.
000:21:52.512 --> 000:22:02.762
If n equals 1, then I get 1 plus 1 over 1 to
000:22:02.762 --> 000:22:09.261
the first power, so this is 2 to the first, which of course is 2.
000:22:09.261 --> 000:22:19.500
What if n is 5: 1 plus 1/5 to the fifth power.
000:22:19.500 --> 000:22:34.750
We can use a calculator to approximate this.
000:22:34.750 --> 000:22:55.822
Okay, let's try n equals 10, and let's just keep five decimal places
000:22:55.822 --> 000:22:57.573
to make this easier to write down.
000:22:57.573 --> 000:23:03.822
This'll be 2.59374 to five decimal places.
000:23:03.822 --> 000:23:31.500
Let's try one more: n equals 100.
000:23:31.500 --> 000:23:35.111
Okay, so what you can see by examining these
000:23:35.111 --> 000:23:37.862
numbers is that the value of the expression is
000:23:37.862 --> 000:23:40.118
getting bigger, the bigger n gets,
000:23:40.118 --> 000:23:44.000
but it's not getting as big as fast.
000:23:44.000 --> 000:23:46.118
In fact, if we were to keep going with
000:23:46.118 --> 000:23:48.869
say a 1000 or 10000 or 1000000, we would
000:23:48.869 --> 000:23:51.630
start to approach one number over here, and
000:23:51.630 --> 000:24:00.133
that number is defined as e. It is approximately 2.718.
000:24:00.133 --> 000:24:02.385
Each of your calculators will have a button
000:24:02.385 --> 000:24:07.000
that will allow you to calculate e to a power. Let's look at that next.
000:24:08.300 --> 000:24:10.141
Alright, so one thing we want to be able to
000:24:10.141 --> 000:24:13.141
do is find e to any real number power, and
000:24:13.141 --> 000:24:15.393
we're going to use the calculator to do that.
000:24:15.393 --> 000:24:18.900
You want to look at your calculator and find e to the x.
000:24:18.900 --> 000:24:22.658
On many calculators it'll be the second function on the ln button.
000:24:22.658 --> 000:24:25.164
If you can't find it, look in your owner's manual.
000:24:25.164 --> 000:24:35.200
Alright, so we want e to the 1.4 to three decimal places.
000:24:35.200 --> 000:24:37.100
So, using a calculator, we're going to raise
000:24:37.100 --> 000:24:48.437
e to that power and round off to three decimal places.
000:24:48.437 --> 000:24:57.000
So, this is approximately 4.055.
000:24:58.200 --> 000:25:02.883
Let's look at the graph of the exponential function with base e.
000:25:02.883 --> 000:25:03.883
The first thing we want to do is sketch the
000:25:03.883 --> 000:25:10.132
graph of y equals e to the x. Well, we just said that
000:25:10.132 --> 000:25:15.200
e is approximately 2.7, okay. So, we're going to use that to draw our graph.
000:25:15.200 --> 000:25:16.979
It's also going to give us an idea of the
000:25:16.979 --> 000:25:19.979
shape of the graph before we even begin: 2.7
000:25:19.979 --> 000:25:23.479
is bigger than 1, so we're going to have a
000:25:23.479 --> 000:25:25.490
graph that looks something like this.
000:25:25.490 --> 000:25:33.248
Let's plot a few extra points. It has to have the point (0,1),
000:25:33.248 --> 000:25:36.998
and it has to have the point (1,e).
000:25:36.998 --> 000:25:46.100
Since e is about 2.7, I'll place the point between 2 and 3 for the y value.
000:25:46.100 --> 000:25:48.999
Keep in mind that basic shape, and there's
000:25:48.999 --> 000:25:52.720
your graph of y equals e to the x.
000:25:52.720 --> 000:25:54.247
The next thing we want to do is use
000:25:54.247 --> 000:25:58.498
transformations to graph y equals e to the x plus 1.
000:25:58.498 --> 000:26:00.300
We only have to worry about one transformation.
000:26:00.300 --> 000:26:02.275
What kind of transformation is it?
000:26:02.275 --> 000:26:06.031
Since the plus 1 is added to the basic
000:26:06.031 --> 000:26:15.000
function e to the x, it tells us we want to move the entire graph up 1 unit.
000:26:15.000 --> 000:26:16.533
So, we're looking at this graph, and we want
000:26:16.533 --> 000:26:22.500
to move everything up 1 unit. The first thing I want to move is the asymptote.
000:26:22.500 --> 000:26:27.554
Use a dotted line for your asymptote, and then move each of the points
000:26:27.554 --> 000:26:35.801
on your graph up 1 unit. So, I want to be careful with this one.
000:26:35.801 --> 000:26:39.500
Right now, the y value is approximately 2.7.
000:26:39.500 --> 000:26:47.000
I want to move it up 1, so that would make it about 3.7.
000:26:47.000 --> 000:26:49.073
Keep in mind the basic shape of the graph,
000:26:49.073 --> 000:26:50.822
and this is our asymptote, so it's never
000:26:50.822 --> 000:26:54.581
going to cross or touch the dotted line.
000:26:54.581 --> 000:26:59.081
This is a graph of y equals e to the x plus 1.
000:26:59.081 --> 000:27:01.082
Now we want to state the domain, the range,
000:27:01.082 --> 000:27:06.200
and the equation of the asymptote.
000:27:06.200 --> 000:27:10.093
The domain is the set of all real numbers as
000:27:10.093 --> 000:27:14.593
it is for any exponential function, like the ones we've studied.
000:27:14.593 --> 000:27:20.093
Its range, we should be able to get that by looking at the graph.
000:27:20.093 --> 000:27:23.092
It never goes below the line y equals 1, so
000:27:23.092 --> 000:27:27.500
that means of y values need to always be bigger than 1.
000:27:27.500 --> 000:27:30.613
In interval notation, it'll look like this.
000:27:30.613 --> 000:27:33.364
Finally, the equation of the asymptote.
000:27:33.364 --> 000:27:37.623
It's a horizontal asymptote: in fact, it's the horizontal line y equals 1,
000:27:37.623 --> 000:27:52.123
so that's the equation. Now it is important to note that
000:27:52.123 --> 000:27:56.641
e is not equal to 2.7. I use that in order to draw my graph, but e
000:27:56.641 --> 000:27:59.639
is really an irrational number, much like pi.
000:27:59.639 --> 000:28:04.140
It's approximately 2.718, but it goes on indefinitely.
000:28:04.140 --> 000:28:06.641
e is also very important because many
000:28:06.641 --> 000:28:08.890
naturally occurring phenomena such as
000:28:08.890 --> 000:28:11.140
compound interest, radioactive decay,
000:28:11.140 --> 000:28:13.890
population growth, and many other things
000:28:13.890 --> 000:28:18.250
have a base e exponential function that describes them.
000:28:20.300 --> 000:28:22.914
Our last two examples are going to show us a
000:28:22.914 --> 000:28:24.666
little bit more about what exponential
000:28:24.666 --> 000:28:26.664
functions are and how they behave and, in
000:28:26.664 --> 000:28:30.918
fact, give us some idea of what it means to grow exponentially.
000:28:30.918 --> 000:28:33.928
For this one, it says determine whether the
000:28:33.928 --> 000:28:36.500
function given in the table is exponential or not.
000:28:36.500 --> 000:28:39.431
If it is exponential, identify the value of
000:28:39.431 --> 000:28:42.400
the base a, and it even gives you a hint:
000:28:42.400 --> 000:28:48.194
look at ratios of consecutive values. So, let's take that hint.
000:28:48.194 --> 000:28:50.194
We want to look at ratios of consecutive
000:28:50.194 --> 000:28:53.300
function values, so let's look at these two.
000:28:53.300 --> 000:28:56.200
The 1 is four times bigger than 1/4.
000:28:56.200 --> 000:28:59.201
Okay, now let's look at the next pair.
000:28:59.201 --> 000:29:03.451
Four is four times bigger than 1. Okay, take the next pair.
000:29:03.451 --> 000:29:06.451
I should point out that they are consecutive
000:29:06.451 --> 000:29:07.951
values because x is increasing by 1 each
000:29:07.951 --> 000:29:11.466
time, so you want to look out for that.
000:29:11.466 --> 000:29:17.972
16 is four times bigger than 4, and 64 is four times bigger than 16.
000:29:17.972 --> 000:29:21.723
So, it's like you're raising 4 to one bigger
000:29:21.723 --> 000:29:26.723
power each time, multiplying by 4 with each
000:29:26.723 --> 000:29:30.485
consecutive increase in x, and that means that
000:29:30.485 --> 000:29:40.800
we have an exponential function with base 4.
000:29:42.500 --> 000:29:45.734
Our last example gives us the graph of an
000:29:45.734 --> 000:29:48.887
exponential function and asks us to write its equation.
000:29:48.887 --> 000:29:50.639
Okay, so let's examine the graph.
000:29:50.639 --> 000:29:54.140
First of all, it is exponential: it has the right shape.
000:29:54.140 --> 000:29:57.390
The base should be bigger than 1, based on the shape.
000:29:57.390 --> 000:30:00.397
We look at the function values that
000:30:00.397 --> 000:30:11.160
we're given: 1/7 when x is negative 1, 1 when x is 0, 7 when x is 1, 49 when x is 2.
000:30:11.160 --> 000:30:16.800
So, those look to me like powers of 7.
000:30:16.800 --> 000:30:22.400
So, let's try it: y equals 7 to the x, and you can double check yourself.
000:30:22.400 --> 000:30:25.670
If you make x negative one, you'll get 1/7;
000:30:25.670 --> 000:30:30.671
if you make x zero, 7 to the 0 is 1, that's
000:30:30.671 --> 000:30:35.671
good; if x is 1, 7 to the first is 7, so you can check yourself.
000:30:35.671 --> 000:30:38.100
You'll need to go back and practice all
000:30:38.100 --> 000:30:39.688
these skills related to exponential
000:30:39.688 --> 000:30:41.438
functions because you'll use them very much
000:30:41.438 --> 000:30:45.000
for the remainder of your mathematical studies.