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Graphing exponential growth & decay

Graphing the exponential functions y=27⋅(⅓)ˣ and y=-30⋅2ˣ.

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  • aqualine tree style avatar for user Taylor Willingham
    Would the asymptote for an exponential function ever be anything but zero?
    (69 votes)
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  • male robot hal style avatar for user Feraru Silviu Marian
    Is there previous lesson about asymptotes, or was it introduced it in this video the first time ?
    (38 votes)
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  • mr pink red style avatar for user Lucio
    he didn't explain the asymptote at all?
    (21 votes)
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  • blobby green style avatar for user jfails029
    How does this relate to the real world
    (7 votes)
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    • aqualine ultimate style avatar for user AD Baker
      JFails,

      Exponential growth comes up a lot in business (investments), in social science (population growth), and in virology (how quickly a virus spreads).

      Exponential decay comes up in audio engineering (sound levels decrease exponentially over longer distance), sports tournaments (if a tournament starts with 32 teams, how many rounds are there until the championship), archaeology(radiocarbon dating), and climate studies (atmospheric pressure decreases exponentially with increased altitude.)
      (16 votes)
  • male robot johnny style avatar for user Jacob Benda
    In other higher level math courses, lines can occasionally cross a horizontal asymptote. Is this possible when graphing exponential functions?
    (9 votes)
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  • blobby green style avatar for user evangelia frantsvog
    how would you graph a function with a negative base like y= -5^x
    (4 votes)
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    • piceratops ultimate style avatar for user Fai
      I'm assuming you mean y=-(5)^x (the positioning of the negative sign is very important). In that case, you'll plot the graph of y=5^x and then reflect it over the y axis.

      In the case where you want to take the exponent of a negative value, it gets a lot more tricky, because sometimes, the function is not defined for negative values. for x=0.5, y=(-5)^x is not defined on the real line (you're taking the square root of a negative number).
      (7 votes)
  • male robot hal style avatar for user mama976
    Is the asymptote always horizontal and on the zero line? Also how do you figure where the asymptote lies. Lastly, what is the asymptote and is this asked or used on the SAT. Whoever answers my questions will get an upvote from me! Please help me out here.
    (3 votes)
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    • mr pink green style avatar for user David Severin
      No, there are vertical and other asymptotes as well. For exponential functions, the basic parent function is y=2^x which has a asymptote at x=0, but if it is shifted up or down by adding a constant (y = 2^x + k), the asymptote also shifts to x=k. I do not know what all is on the SAT, but if you have a rational function whose parent function is y = 1/x, you have a horizontal asymptote at x=0 and a vertical asymptote at y=0. These both can be shifted by adding (h,k) so that y = 1/(x-h) + k.
      (7 votes)
  • male robot hal style avatar for user WAJIH RIZVI
    Can an exponential function have an asymptote that is vertical?
    (4 votes)
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  • mr pants teal style avatar for user Emmery
    How do I find where the asymptote goes?
    Thanks! :)
    (2 votes)
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  • female robot amelia style avatar for user Jahara Campbell
    What would you do if the values were reversed? For example, what would the solution be if you graphed the exponential functions as y=1/3*(27)^x?
    (3 votes)
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Video transcript

- [Voiceoer] This is from the graph basic exponential functions on Khan Academy. They asked us graph the following exponential function. And they give us the function, h of x is equal to 27 times 1/3 to the x. So our initial value is 27 and 1/3 is our common ratio. It's written in a standard exponential form. And it gives us some graphing tool where we can define these two points and we can also define a horizontal asymptote to construct our function. And these three things are enough to define to graph an exponential if we know that it is an exponential function. So let's think about it a little bit. So the easiest thing that I could think of is, well, let's think about its initial value. Its initial value is going to be x equal to zero. X equal zeros, 1/3 to the zero power is just one and so you're just left with 27 times one or just 27. That's what we call this number here when you've written in this form. You call this the initial value. So when x is equal to zero, h of x is equal to 27. Now we're graphing y=h(x). So now let's graph another point. So let's think about it a little bit. When x is equal to one, when x is equal to one, what is h of x? It's gonna be 1/3 to the first power which is just 1/3. And so 1/3 times 27 is gonna be nine. So when x is one, h of one is nine. And we can verify. And now let's just think about, let's think about the asymptote. So what's gonna happen here when x becomes really, really, really, really, really, really, really big? If I take 1/3 to like a really large exponent to, say to the 10th power or to the 100th power or to the 1000th power, this thing right over here is going to start approaching zero as x becomes much, much, much, much larger. And so something that is approaching zero times 27, well, that's going to approach zero as well. So we're gonna have a horizontal asymptote at zero. And you can verify that this works for more than just a two points we talk about. When x is equal to, this is telling us that the graph, y equals h of x goes to the point two comma three. So h of two should be equal to three. And you can verify that that is indeed the case. If x is two, 1/3 squared is nine. Oh sorry, 1/3 squared is 1/9 times 27 is three. And we see that right over here. When x is two, h of two is three. So I feel pretty good about that. Let's do another one of this. So graph the following exponential function. So same logic. When x is zero, the g of zero is just gonna boiled out to that initial value. And so you scroll down. Initial value is negative 30. And so let's think about when x is equal to one. When x is equal to one, two to the first power is just two. And so two times negative 30 is negative 60. So when x is equal to one, the value of the graph is negative 60. Now let's think about this asymptote over that, it should sit. So let's think about what happens when x becomes really, really, really, really, really, really negative. So when x is really negative, so two to the negative one power is 1/2. Two to the negative two is 1/4. Two to the negative three is 1/8. As it gets larger and larger negative or higher magnitude negative values or in other words, as x becomes more and more and more negative, two to that power is going to approach zero. And so negative 30 times something approaching zero is going to approach zero. So this asymptote is in the right place, a horizontal asymptote as x approaches negative infinity. As we move further and further to the left, the value of a function is going to approach zero. Now we can see it kind of approaches zero from below. We can see they approaches below because we already looked at the initial value and we used that common ratio to find one other point. Hopefully, you found that interesting.