Video Transcript
In this video, we’re gonna be answering some questions about exponential graphs. If you’re not already familiar with how exponential graphs and functions work, then you might like to consider watching our introduction to exponential graphs to find out more. Which of the following functions is represented by this graph? And then we’ve got four functions; 𝑦 equals one to the power of 𝑥, 𝑦 equals nought point five to the power of negative 𝑥, 𝑦 equals negative nought point five to the negative 𝑥, and 𝑦 equals nought point five to the power of 𝑥.
Well the first thing to think is that all of our possible answers there are exponential functions. And it does look like we’ve got an exponential function, but it’s showing exponential decay rather than exponential growth. Now as our 𝑥-coordinate is increasing and becoming positive, then the corresponding 𝑦-coordinates are getting smaller and smaller and smaller down to zero. So they’re getting close to zero, but not equal to zero. And they’re certainly not negative. And as the 𝑦-coordinates are getting smaller, obviously the curve is getting closer and closer and closer to the 𝑥-axis. Now the general format of the exponential function is 𝑦 is equal to some constant, 𝑎, times some constant, 𝑏, to the power of 𝑥. So our variable is in the exponent.
Now looking at the format of that, there are two different ways that we can get exponential decay. Either the 𝑏-value needs to be between zero and one. So we’ve got a fractional number, which were gonna multiply by itself lots of times. So for example, if 𝑏 was a half when 𝑥 is getting large and positive, we’re multiplying half by itself lots of times, and therefore, that’s getting smaller and smaller and closer to zero. Or we can have a 𝑏-value which is greater than one, but the exponent is negative 𝑦. So we’ve gotta say 𝑏 to the minus 𝑥. For example, three to the negative 𝑥, then as 𝑥 gets bigger and bigger and bigger and more positive, three to the negative 𝑥 means we’ve got one over three times itself lots and lots of times, and therefore we’re getting a very small number close to zero.
Now straight away looking at our four options, the only one that follows one of those formats is d, so I suspect that’s gonna be the answer. But let’s go through all of these and explore the other possibilities first. Well for a, we’ve got 𝑦 equals one to the power of 𝑥. That’s just one times itself lots of times. The answer is always gonna be one. So 𝑦 equals one to the 𝑥 is gonna be a straight line that looks like that. So we know that that can’t possibly be our answer. Now for option b, we’ve got nought point five to the power of negative 𝑥, so the 𝑏-value is nought point five. And it’s one lot of that, so the 𝑎-value is one. Well with those values of 𝑎 and 𝑏, we can see that the curve does in fact cut the 𝑦-axis at one, so it would match the situation here; 𝑎 equals one does cut the 𝑦-axis at one. So now we need to go and investigate other factors.
One useful method on a curve like this where we can see the coordinates quite clearly, we know that when 𝑥 is equal to one, the corresponding 𝑦-coordinate is nought point five. So substituting those values in here when 𝑥 is one, 𝑦 is nought point five, we’ll be saying that nought point five is equal to nought point five to the power of negative one. Well a half nought point five to the power of negative one is one over a half which is two, so that wouldn’t be true. So it can’t be that either. Now for c, we can rearrange this equation slightly. That means negative one times nought point five to the power of negative 𝑥. So 𝑎 is equal to negative one. The curve doesn’t cut the 𝑦-axis at negative one, so we know that that’s wrong. And again we could — no point in doing this really, because we’ve already proved it’s not that curve — but we could check this point here. When 𝑥 is one, 𝑦 is nought point five. Well negative a half to the power of negative one, to the power of negative one means we flip that fraction, so negative half becomes negative two. Well negative two isn’t equal to nought point five, so again it’s not that one.
So we strongly suspect that the answer is d now, so let’s just check it. Well we can rearrange it to be one times nought point five to the power of 𝑥. So 𝑎 is one, and yep it does cut the 𝑦-axis at one, so that matches. And then when 𝑥 is one, 𝑦 is nought point five. So let’s put those values in and that will mean that nought point five is equal to nought point five to the power of one, which it does. So our answer is d, 𝑦 equals nought point five to the power of 𝑥.
Now our next question is which of the following functions is represented by this graph. And we’ve got four options: a) 𝑦 is equal to a half to the power of negative 𝑥; b) 𝑦 is equal to negative two to the power of negative 𝑥; c) 𝑦 is equal to negative half to the power of 𝑥; and d) 𝑦 is equal to negative two to the power of 𝑥. Well the first thing to notice is that we’ve got exponential growth, but this been reflected in the 𝑥-axis. So as the 𝑥-coordinates get larger and larger than, the 𝑦-coordinates get further and further away from zero, but in fact they’re going negative rather than positive. Now the scale at which we’ve been given this graph enables us to determine a couple of things quite quickly. It cuts the 𝑦-axis at negative one, so when 𝑥 is equal to zero, then 𝑦 is equal to negative one. We can also see that when 𝑥 is one, the 𝑦-coordinate corresponding to that is negative two. And when 𝑥 is negative one, then the corresponding 𝑦-coordinate is negative a half.
So the easiest way of tackling this question is just to try out those ordered pairs in our equations and see which ones fit, which ones don’t. So in a, let’s try 𝑥 equals zero and see what the corresponding 𝑦-coordinate comes out to be. Well when 𝑥 is zero, we’ve got 𝑦 is equal to a half to the negative zero, which is just the same as a half to the zero. And anything to the power of zero is just one. So where 𝑥 is zero, 𝑦 is one. Well that we’ve fallen in the first hurdle there, we were hoping for 𝑦 to be negative one, so it’s not a. Let’s try b. And in equation b when 𝑥 is zero, we’ve got 𝑦 is equal to negative two to the power of negative zero. Well that’s the same as negative two to the power of zero, and two to the power of zero is just one. So this is negative one.
Great! So it passes that test. Let’s try the next one then. When 𝑥 is one, we want 𝑦 to be equal to negative two. Well when 𝑥 is one, 𝑦 is negative of two to the power of negative one. Well two to the negative one is one over two, so that’s a half. So this is negative a half. But we wanted it to be negative two, so it fails the test.
Let’s have a look at c then. Try 𝑥 equals zero. Well then 𝑦 is equal to the negative of half to the power of zero. Well half to the power of zero, anything to the power of zero is one, so that is negative of one. That’s negative one. That’s good. That’s what we wanted it to be, so we’re gonna have to try something else now. So just because it cuts the 𝑦-axis in the right place doesn’t mean to say that it’s the same curve in all the places. So let’s try when 𝑥 is equal to one. Well when 𝑥 is one, we’ve got 𝑦 is equal to the negative of a half to the power of one. Well a half to the power of one is just a half and the negative of that is negative a half. But when 𝑥 was equal to one, we wanted the 𝑦-coordinate to be negative two, so that is also wrong. Let’s hope it’s d then. Let’s try 𝑥 equals zero.
When 𝑥 is zero, then 𝑦 is equal to negative two to the power of zero. The two to the power of zero is one, so this is negative one. Great! That looks like that works, so now let’s try 𝑥 equals one. And 𝑦 is equal to negative two to the power of one. Two to the power of one is just two, so the negative of two is negative two. And yep, that’s what we wanted. So now let’s just try 𝑥 is equal to negative one. And in that case, 𝑦 would be equal to the negative of two to the negative one. Well two to the negative one is one over two, so that’s a half. So that gives us negative a half, and that is in fact what we were looking for. So we validated all those three points. So we haven’t proved that that’s the exact curve, but we’ve certainly validated it and we’ve certainly disproved all the others. So this looks like the right answer.
Now we could try some more points or we could just think that, you know, two to the power of 𝑥 would be exponential growth. You think about it, two to the zero, two to the one, two to the two, two to the three, and so on. And then we’re taking the negative of all those 𝑦-coordinates, so we’re reflecting that in the 𝑥-axis. So it kind of matches on lots of fronts. So our answer is d. 𝑦 is equal to negative two to the 𝑥.
Now let’s consider this one, where would the graph of 𝑦 equals five to the power of 𝑥 intersect the 𝑦-axis? Well intersecting the 𝑦-axis means cutting the 𝑦-axis, and that means that it’s got an 𝑥-coordinate of zero. And when 𝑥 equals zero, then 𝑦 is equal to five to the power of 𝑥, so five to the power of zero. Five the power of zero is one. So our answer is that it would intersect the 𝑦-axis at zero, one.
And here’s another one, where would the graph of 𝑦 equals seven point one times six to the power of negative 𝑥 intersect the 𝑦-axis? Well again all points on the 𝑦-axis, have got an 𝑥-coordinate of zero, so this curve’s gonna cut the 𝑦-axis when 𝑥 is equal to zero. So let’s put that into our equation. So when 𝑥 is equal to zero, then 𝑦 would be equal to seven point one times six to the power of negative zero. Well negative zero is just the same as zero, so that’s six to the power of zero. Well anything to the power of zero, as long so it’s not zero, is one. So this becomes 𝑦 is equal to seven point one times one. Seven point one times one is clearly seven point one. So our answer is that it cuts the 𝑦-axis at zero, seven point one.
The graph of the function 𝑦 equals 𝑏 to the power of 𝑥 passes through the point five, one. Find the value of 𝑏. Well this means that if it’s passing through this point five, one, it means that when the 𝑥-coordinate is equal to five, then the corresponding 𝑦-coordinate is equal to one. So we can put those values for 𝑥 and 𝑦 into our equation and solve for 𝑏. So if 𝑦 equals 𝑏 to the power of 𝑥 when 𝑥 is five, that means it’s gonna be 𝑏 to the power of five. Then 𝑦 is one, so that whole thing is gonna be equal to one. I’m gonna take the fifth root of both Sides so that I can work out what just 𝑏 is. So on the left-hand side, I’m gonna get the fifth root of one. And the fifth root of 𝑏 to the power of five is just 𝑏, which was obviously the point of taking fifth roots. Now the fifth root of one is just one, because one times itself, five times is one. So we found out that 𝑏 is equal to one.
The graph of 𝑦 equals 𝑚 to the power of 𝑥 passes through the point three, twenty-seven. Find the value of 𝑚. And again this tells us that when 𝑥 is equal to three, then the corresponding 𝑦-coordinate would be equal to twenty-seven. So again we just need to plug those values of 𝑥 and 𝑦 into our equation and solve for 𝑚. So twenty-seven is equal to 𝑚 to the power of three. And this time I’m gonna take cube roots of both sides so that I can find out what just 𝑚 is. So on the left-hand side, that gives me the cube root of twenty-seven; and on the right-hand side, the cube root of 𝑚 cubed is just 𝑚. And the cube root of twenty-seven is three, so 𝑚 is equal to three.
