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Analyzing the features of exponential graphs through the example of y=5ˣ. Created by Sal Khan and Monterey Institute for Technology and Education.
Video transcript
We're asked to graph y is equal to 5 to the xth 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 lets try some negative and 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 25th. 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 1st power, or just 1 5th. Now let's think about when x is equal to 0. Then y is going to be equal to 5 to the 0th power. Which we know anything to the 0th power is going to be equal to, 1. So this is going to be equal to, 1. And then, finally we have, actually lets try a couple more points here. Lets try out. We extend this table a little bit further. Lets try out x is equal to, 1. Then y is 5 to the 1st power which is just equal to 5. 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 2nd power which is just equal to 25. And now we can plot it to see how this actually looks. To see how it actually looks when we get some graph paper going here. My xs go as low as negative 2. As high as positive 2. And then my ys go all the way from, 1 25th all the way to 25. So let me. So I have positive values over here. So let me draw it like this. So, this can be my x-axis. That could be my x-axis and then lets make this my y-axis. Draw it as neatly as I can. So, lets make that my, y-axis and my x values. This could be, negative 2. I should make my y-axis keep going. So that's y, this is x, that's a negative 2, that's negative 1. This is 0. That is 1, and that is, positive 2. And let's plot the points. X is -2, y is 1 25th. 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. You have to do a little bit smaller than that too. So this is gonna be 5, 10, 15, 20, and then 25 would be right where I wrote the y give or take. So now lets plot them. Negative 2, 1 25th. Negative 2 1 25th. 1 is gonna be like there so 1 25th is gonna be really, really close to the x-axis. That's about 1 25th, so that is. Negative 2, 1 25th. It's not gonna be on the x-axis, 1 25th is obviously greater than 0. It's gonna be really, really, really, really close. Now, let's do this point here in orange. Negative 1, 1 5th. Negative 1 5th. 1 5th on this scale is still pretty, pretty close. It's. Pretty close. So that right over there is negative 1, 1 5th and now in blue, we have 0,1, 0,1 is gonna be right about there if this is 2 and one half it looks about, right from 1. And then we have 1,5. 1, 5 puts us right over there. And then finally we have 2, 25. When x is 2, y is 25. 2, 25. Puts us right, about there. And so I think you see what happens with this function, with this graph. It starts off, the further negative, in the negative, the further in the negative direction we go. 5 to, every increasing negative powers to gets us closer to closer to 0 but never quite. So, we're leaving 0 getting. For slightly further, further, further, from 0. Right at the, right at the y-axis we have y equal 1, and then once, once the exponent over here or right at x is equal to 0, we have y is equal to 1. And then what, 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 kind of this, sometime 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 get closer, and closer, and closer to 0 without quite getting to 0. So, 5 to the negative billionth power. It's still not gonna get you to 0, but it's gonna get you pretty darn close to 0. But obviously if you go to 5 to the positive billionth power you're gonna get a super, huge number, cuz this thing is just going to keep, sky rocketing up like that. So let me just draw the whole curve just 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, is once I get into the positive xs, then I start really, really shooting up.