Exponential function graph

we're asked to graph y is equal to 5 to the X 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 125th 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 0th power which we know anything to the zeroth 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 more points here let's try out 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 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 to see how would actually look so let me get some graph paper going here my X's goes low as negative 2 is high as positive 2 and then my Y's go all the way from 125th all the way to 25 so let me 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 draw it as neatly as I can so let's make that my y-axis and my x-values this could be negative 2x you make my y-axis keep going so that's why this is X that's a negative 2 it's negative 1 that's 0 that is 1 and that is positive 2 let's plot the points X is negative 2 y is 125th actually let me point 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 which you have to do a little bit smaller than that 2 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 25th negative 2 125th 1 is going to be like there so 125th is going to be really really close to the x-axis that's about 125th so that is negative 2 125th it's not going to be on the x axis 125th 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 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 it starts off the further negative in the negative x the further in the negative direction we go 5 2 ever-increasing negative powers gets us closer and closer to zero but never quite so we're leaving 0 getting slightly further further further from zero right at the right at the y-axis we have equal one and then once once the exponent over here or right at X is equal to zero we have Y is equal to one and then one at once X starts increasing beyond zero then we start seeing what the exponential is good at which is just this very rapid increase I'm grievin we 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 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 you get closer and closer and closer to zero without quite getting to zero so 5 to the negative billionth power it's still not going to get you to zero but it's going to get you pretty darn close to zero but obviously if you go to five to the positive billionth power you're gonna 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 make sure you see it over here I'm not actually on zero all the way the way I drew it it might look like that I'm slightly above zero I'm increasing above that increasing about that as once I get into the positive X's then I start really really shooting up