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### Course: Algebra I (2018 edition) > Unit 13

Lesson 3: Graphs of exponential growth# Graphs of exponential growth

Identifying which graph represents a given exponential function.

## Want to join the conversation?

- can somebody please tell me what does f(x) mean(8 votes)
- Okay, so you know what "y = 3x + 4" means right?

Well, when we input an x into that equation, we map out a y value. As we get into more advanced math, we will start using "f(x) = 3x + 4" instead of "y = 3x + 4". But they are essentially the same thing; f(x) is a function where if we input any non-restricted x value we will map out a "y" value. Thus "y = f(x)". One convenient use of "f(x)" is that we can use separate equations/functions and not confuse ourselves. e.g.:

f(x) = 3x + 4

g(x) = (1/2)x - 2

If we used y, then we could get confused by whether or not we were talking about the same equation/function. So using function notation removes the confusion there.

Hope this helps,

- Convenient Colleague(25 votes)

- explain to me why a negative power is always a fraction? and Why Sal drew curved lines between the points.(4 votes)
- As an example, going backwards from 2^3 = 8, divide both sides by 2 gives 2^2 = 4, 2^1 = 2, 2^0 = 1. When we keep going, 2^-1=.5 = 1/2, 2^-2 = .25 = 1/4, etc. However, a negative power is not always a fraction, it is a reciprocator. So 1/(2^-2) = 2^2 = 4.(3 votes)

- Is it correct that in this example the x-intercept doesn't exist since the graph never touches the x-axis?

Could anyone give me an example of an exponential function which would have an x-intercept if we graphed it?

I'm thinking of something along the lines of*f(x)=m*(n^x)-c*(though would we still call it an exponential function or is it more like a "combined" sort of thing?), but I'm interested whether there is a bare-bones exponential function of the form f(x)=m*n^x without adding or subtracting anything(4 votes)- An standard exponential function will never cross the x-axis. It will always approach but never touch the x-axis as it approaches negative infinity. In more advanced terms, the x-axis is a horizontal asymptote of the exponential function. For an example of this, see the video below from the time4:00to4:15(the time stamps are not for the video above but they are video below) and4:50to5:07.

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:graphs-of-exponential-growth/v/graphing-exponential-functions(3 votes)

- how do you graph an exponential function with a table??(0 votes)
- You pick values for X and calculate the corresponding Y value just like Sal does in the video.

You should then have a list of ordered pairs (x, y).

Graph them.(12 votes)

- So I have a question my problem is: y=-2(1/6)^x how would I do this since it is a fraction?(1 vote)
- think about what happens when you have 2^x. At x=0, you get 1 and as x gets bigger, it increases exponentially (1,2)(2,4)(3,8). On the other side, as x goes negative, it turns to fractions (-1,1/2)(-2,1/4), etc. Fractions would do the opposite such as (1/2)^x. 0 would still give 1, but to the right you get (1,1/2)(2,1/4) etc. and to the left you would get (1/2)^-1 = 2^1, so (-1,2)(-2,4)(-3,8) etc.

Your problem has much more than a fractional base, you have a scale factor of -1 along with the fractional base. The negative reflects it across the x axis, the 2 vertically stretches the function, and the base of 1/6 has it approaching negative infinity as you go to the left and 0 as you go to the right. If x=0, you would be at (0,-2), at -1 it would be -2(6)=-12, at -2 it would be -2(6)2=-72, etc. to the right, at 1, it would be -2*1/6 = -1/3, at 2 it would be -2*1/6^2=-1/18, etc.(5 votes)

- Where did you go to college?(3 votes)
- Is it right to say that the exponential and linear functions are geometric and arithmetic sequences respectively?(3 votes)
- You can say that, however technically speaking, the term numbers of sequences (inputs) have to have positive, integer, values (some people use zeroth terms so the term number can be non-zero values). An exponential and linear function can have negative, decimal inputs, so in rigorous mathematical language, you can't say that, but informally speaking, the concepts is very, very similar.(2 votes)

- question why does anything to the power of 0=1?(1 vote)
- See this lesson: https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-arithmetic-operations/cc-6th-exponents/v/the-zeroth-power

FYI - You can find topics quickly by using the search bar at the top of all KA screens.(5 votes)

- The word exponential growth and linear growth are frequently used in the media. People toss these words around wrongly. With a very easy to follow numerical example ,what exactly does it mean to grow exponentially?(1 vote)
- In a mathematical sense I believe that to "grow exponentially" simply means that you are modeling growth (such as a population) with an exponential function (usually of the form y=a^x). The reason we use the term exponential growth as opposed to linear growth in media is because we are comparing a growth rate to the behavior of an exponential function. So for example something with a linear growth rate will grow at a steady pace while something that has an exponential growth rate is increasing extremely rapidly after only a small amount of time. We can picture this behavior using the graph of an exponential function, say y=2^x, for every increase in x, y grows faster and faster (when x=1, y=2, when x=2, y=4, when x=4, y=16 etc.) instead of at a constant rate.

Hope this helps! :)(3 votes)

- what is the correlation between a function's rate/slope and its common ratio?(2 votes)
- The b value would be the common/constant ratio. Every time the x value moves up one, the y-value would be multiplied by the b value (the opposite is true when the x value moves down one). Thus, given the equation, you simply have to start at the y-intercept (the a value) and multily/divide by the b value to graph it.(1 vote)

## Video transcript

- [Instructor] Alright,
we are asked to choose the graph of the function. And the function is f(x) is equal to two, times three to the x and we have three choices here. So, pause this video and
see if you can determine which of these three graphs actually is the graph of f(x). Let's work through this together. So, whenever I have a function like this, which is an exponential function, because I'm taking some number and I'm multiplying it by some
other number to some power. So, that tells me that I'm
dealing with an exponential. So, I like to think about two things. What happens when x equals zero? What is the value of our function? Well, when you just look at this function, this would be two,
times three to the zero. Which is equal to, three
to the zero is one. It's equal to two. So, one way to think about it. In the graph of y is equal to f(x), when x is equal to
zero, y is equal to two. Or another way to think about it is this value in exponential function, sometimes called the initial value, if you were thinking of the x-axis. Instead of the x-axis, you're
thinking about the time axis or the t-axis. That's why it's sometimes
called the initial value. But the y-intercept is
gonna be described by that when you have a function of this form. And you saw it right over there, f(0). Three to the zero's one. You're just left with the two. So, which of these have
a y-intercept of two? Well, here, the
y-intercept looks like one. Here, the y-intercept looks like three. Here, the y-intercept is two. So, just through elimination
through that alone, we can feel pretty good that this third graph is probably the choice. But let's keep on analyzing it to feel even better about it. And so, we have the skills for really any exponential function
that we might run into. Well, the other thing to realize. This number, three, is often referred to as a common ratio. And that's because every
time you increase x by one, you're gonna be taking
three to a one higher power. Or you're essentially gonna
be multiplying by three again. So, for example, f(1) is going to be equal to two, times three to the one. Two, times three to the
one or two times three, which is equal to six. So, from f(0) to f(1), you essentially have to multiply by three. And you keep multiplying by three. f(2) you're gonna multiply by three again. It's gonna be two, times three squared, which is equal to 18. And so, once again, when
I increased my x by one, I'm multiplying the value
of my function by three. So, let's just see which of these do this. This one we said has
the wrong y-intercept, but, as we go from x equals
zero to x equals one, we are going from one to three. And then, we are going from three till looks like pretty close to nine. So, it does look like this does have a common ratio of three. It just does have a different y-intercept than the function we care about. This looks like the graph f(x) is equal to just
one, times 3 to the x. Here, we're starting at three. And then, when x equals one, it looks like we are doubling
every time x increases by one. So, this looks like the
graph of y is equal to... I have what we could
call our initial value, our y-intercept, three. And, if we're doubling every
time, we increase by one. Three, times two to the x. That's this graph here. As I said, this first graph looks like y is equal to one, times three to the x. We are tripling every time. One, times three to the x. Or we could just say y is
equal to three to the x. Now, this one here better work, 'cause we already picked
it as our solution. So, let's see if that's actually the case. So, as we increase by one,
we should multiply by three. So, two times three is, indeed, six. And then, when you
increase by another one, we should go to 18. And that's kind of off the charts here, but it does seem reasonable to see that we are multiplying by three every time. And you could also go the other way. If you're going down by one, you should be dividing by three. So, two divided by three, this does look pretty close to 2/3. So, we should feel very
good about our third choice.