Graphs of exponential growth
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.