Analyzing the features of exponential graphs through the example of y=5ˣ. Created by Sal Khan and Monterey Institute for Technology and Education.
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- Can anyone explain to me why a negative power is always a fraction?(15 votes)
- Technically, an exponent expresses multiplication, and is shown with a positive number. The opposite of a positive number is (obviously) a negative number, so in keeping with the rules of exponents, this must somehow be the "opposite" of multiplication, which happens to be division. Therefore, a negative exponent is always a division (written as a fraction).
That's the reason.(25 votes)
- whats a slope?(0 votes)
- these answer above are all right.but there's another idea about slope.slope of a line can be expressed as a trigonometric function.it's the tangent of the angle formed between the x-axis and the line itself.(8 votes)
- Why Sal drew curved lines between the points. Why not straight lines?(2 votes)
- Exponential functions do not change in a constant manner. Linear equations have a constant slope. With exponential equations, the change accelerates as the exponent increases. See this video: https://www.khanacademy.org/math/algebra/introduction-to-exponential-functions/exponential-vs-linear-growth/v/exponential-vs-linear-growth(4 votes)
- So no matter what, the graph can't go below the "X-Axis"?(2 votes)
- If a is a positive number, then a^x is greater than 0 for all x. So, the graph of a^x will never intersect, nor go below the x-axis.(2 votes)
- how do you do negative exponents?(1 vote)
- The first step is to flip them for example
5^-2 would flip the -2 and because all numbers are technically fractions (5=5/ 12=12/1 this applies to any number) you would just flip it so instead of having -2 (or -2/1) you would have 1/2. so when rewritten it would be 5^1/2 then you can plug into your cauclour(1 vote)
- Don't Positive Exponential Functions always rise upward from the x-axis while the Negative Exponential Function slides downward to the the x-axis. In the Positive function, both x and y values increase I presume, and in the negative the x value increases, while the y value decreases?(0 votes)
- What is the difference between exponents and indices?(2 votes)
- T Ross,
Exponents are notations that indicate a base number is raised to a power or multiplied by itself a given number of times. In writing or word processing programs that allow it, exponents are written as superscript(above the base number). In a plain text editor (like this one), exponents are noted using the *^* symbol.
2^3 means 2 multiplied together 3 times: 2*2*2 = 8
x^3 means x multiplied together 4 times: x*x*x*x
Indices are a notation that indicates the position of an element in a sequence, array, or matrix. In a word processing program that allows it, indices are shown in subscript (below the name or variable assigned to the sequence). In a plain text editor, indices are indicated using the _. In the example in the video, Sal uses the sequence of numbers -2, -1, 0, 1, 2 for the x values. If you were to call this sequence X, then
X_1 = -2
X_2 = -1
X_3 = 0
X_4 = 1
X_4 = 2(2 votes)
- sorry this question stupid but i really suck at math so where does the two in the equation come from?(2 votes)
If you are talking about the 2 in the column labeled x, that is a value that Sal selected. He selected all of the x values and used them to calculate the y values.(2 votes)
- why does any number to the 0 power always become a 1? why not 0?(2 votes)
- As the saying goes, "Do the Math!"
At this stage you may not know all you need to with regard to the properties of exponents.
Here is one explanation that requires knowing that (x^a)/(x^b)= x^(a-b)
You know that, for example, 5/5=1, correct? It is because the numerator and denominator are equal.
Suppose you had (5^6)/(5^6). Since the numerator and denominator are equal, this is also equal to 1.
Now, using the exponential property that (x^a)/(x^b)= x^(a-b), we have
(5^6)/(5^6) = 5^(6-6) = 5^0.
And since (5^6)/(5^6) = 1 and (5^6)/(5^6) = 5^(6-6) that means 5^0 = 1 as well.
You will know lots more about exponential function when you finish this course!(2 votes)
- At4:06, the video said that the smaller the negative exponent we put, the more closely we will get to zero, but not quite to zero.
So, will we ever able to reach zero on the number line?(2 votes)
- Not with a "normal" exponential function because 0 is a horizontal asymptote. We can shift the exponential function down by subtracting a number at the end such as y = a(b)^x - 3, and this shifts the asymptote down 3 which gives us a x intercept, but then it will get really close to -3 without ever reaching it.
There is an old conundrum that if you are 10 feet from a wall and you go 1/2 the way there every minute, will you ever reach the wall? The theoretical answer is no because you just keep dividing a number by 2, but practically you quickly cannot measure what 1/2 of the way is.(1 vote)
We're asked to graph y is equal to 5 to the x-th power. And we'll just do this the most basic way. We'll just try out some values for x and see what we get for y. And then we'll plot those coordinates. So let's try some negative and some positive values. And I'll try to center them around 0. So this will be my x values. This will be my y values. Let's start first with something reasonably negative but not too negative. So let's say we start with x is equal to negative 2. Then y is equal to 5 to the x power, or 5 to the negative 2 power, which we know is the same thing as 1 over 5 to the positive 2 power, which is just 1/25. Now let's try another value. What happens when x is equal to negative 1? Then y is 5 to the negative 1 power, which is the same thing as 1 over 5 to the first power, or just 1/5. Now let's think about when x is equal to 0. Then y is going to be equal to 5 to the 0-th power, which we know anything to the 0-th power is going to be equal to 1. So this is going to be equal to 1. And then finally, we have-- well, actually, let's try a couple of more points here. Let me extend this table a little bit further. Let's try out x is equal to 1. Then y is 5 to the first power, which is just equal to 5. And let's do one last value over here. Let's see what happens when x is equal to 2. Then y is 5 squared, 5 to the second power, which is just equal to 25. And now we can plot it to see how this actually looks. So let me get some graph paper going here. My x's go as low as negative 2, as high as positive 2. And then my y's go all the way from 1/25 all the way to 25. So I have positive values over here. So let me draw it like this. So this could be my x-axis. That could be my x-axis. And then let's make this my y-axis. I'll draw it as neatly as I can. So let's make that my y-axis. And my x values, this could be negative 2. Actually, make my y-axis keep going. So that's y. This is x. That's a negative 2. That's negative 1. That's 0. That is 1. And that is positive 2. And let's plot the points. x is negative 2. y is 1/25. Actually, let me make the scale on the y-axis. So let's make this. So we're going to go all the way to 25. So let's say that this is 5. Actually, I have to do it a little bit smaller than that, too. So this is going to be 5, 10, 15, 20. And then 25 would be right where I wrote the y, give or take. So now let's plot them. Negative 2, 1/25. 1 is going to be like there. So 1/25 is going to be really, really close to the x-axis. That's about 1/25. So that is negative 2, 1/25. It's not going to be on the x-axis. 1/25 is obviously greater than 0. It's going to be really, really, really, really, close. Now let's do this point here in orange, negative 1, 1/5. Negative 1/5-- 1/5 on this scale is still pretty close. It's pretty close. So that right over there is negative 1, 1/5. And now in blue, we have 0 comma 1. 0 comma 1 is going to be right about there. If this is 2 and 1/2, that looks about right for 1. And then we have 1 comma 5. 1 comma 5 puts us right over there. And then finally, we have 2 comma 25. When x is 2, y is 25. 2 comma 25 puts us right about there. And so I think you see what happens with this function, with this graph. The further in the negative direction we go, 5 to ever-increasing negative powers gets closer and closer to 0, but never quite. So we're leaving 0, getting slightly further, further, further from 0. Right at the y-axis, we have y equal 1. Right at x is equal to 0, we have y is equal to 1. And then once x starts increasing beyond 0, then we start seeing what the exponential is good at, which is just this very rapid increase. Some people would call it an exponential increase, which is obviously the case right over here. So then if I just keep this curve going, you see it's just going on this sometimes called a hockey stick. It just keeps on going up like this at a super fast rate, ever-increasing rate. So you could keep going forever to the left, and you'd get closer and closer and closer to 0 without quite getting to 0. So 5 to the negative billionth power is still not going to get you to 0, but it's going to get you pretty darn close to 0. But obviously, if you go to 5 to the positive billionth power, you're going to get to a super huge number because this thing is just going to keep skyrocketing up like that. So let me just draw the whole curve, just to make sure you see it. Over here, I'm not actually on 0, although the way I drew it, it might look like that. I'm slightly above 0. I'm increasing above that, increasing above that. And once I get into the positive x's, then I start really, really shooting up.