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In this video, we’re gonna be answering some questions about exponential graphs.
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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.
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Which of the following functions is represented by this graph?
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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.
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Well the first thing to think is that all of our possible answers there are exponential functions.
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And it does look like we’ve got an exponential function, but it’s showing exponential decay rather than exponential growth.
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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.
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So they’re getting close to zero, but not equal to zero.
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And they’re certainly not negative.
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And as the 𝑦-coordinates are getting smaller, obviously the curve is getting closer and closer and closer to the 𝑥-axis.
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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.
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So our variable is in the exponent.
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Now looking at the format of that, there are two different ways that we can get exponential decay.
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Either the 𝑏-value needs to be between zero and one.
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So we’ve got a fractional number, which were gonna multiply by itself lots of times.
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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.
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Or we can have a 𝑏-value which is greater than one, but the exponent is negative 𝑦.
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So we’ve gotta say 𝑏 to the minus 𝑥.
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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.
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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.
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But let’s go through all of these and explore the other possibilities first.
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Well for a, we’ve got 𝑦 equals one to the power of 𝑥.
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That’s just one times itself lots of times.
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The answer is always gonna be one.
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So 𝑦 equals one to the 𝑥 is gonna be a straight line that looks like that.
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So we know that that can’t possibly be our answer.
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Now for option b, we’ve got nought point five to the power of negative 𝑥, so the 𝑏-value is nought point five.
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And it’s one lot of that, so the 𝑎-value is one.
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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.
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So now we need to go and investigate other factors.
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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.
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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.
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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.
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So it can’t be that either.
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Now for c, we can rearrange this equation slightly.
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That means negative one times nought point five to the power of negative 𝑥.
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So 𝑎 is equal to negative one.
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The curve doesn’t cut the 𝑦-axis at negative one, so we know that that’s wrong.
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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.
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When 𝑥 is one, 𝑦 is nought point five.
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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.
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Well negative two isn’t equal to nought point five, so again it’s not that one.
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So we strongly suspect that the answer is d now, so let’s just check it.
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Well we can rearrange it to be one times nought point five to the power of 𝑥.
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So 𝑎 is one, and yep it does cut the 𝑦-axis at one, so that matches.
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And then when 𝑥 is one, 𝑦 is nought point five.
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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.
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So our answer is d, 𝑦 equals nought point five to the power of 𝑥.
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Now our next question is which of the following functions is represented by this graph.
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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.
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Well the first thing to notice is that we’ve got exponential growth, but this been reflected in the 𝑥-axis.
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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.
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Now the scale at which we’ve been given this graph enables us to determine a couple of things quite quickly.
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It cuts the 𝑦-axis at negative one, so when 𝑥 is equal to zero, then 𝑦 is equal to negative one.
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We can also see that when 𝑥 is one, the 𝑦-coordinate corresponding to that is negative two.
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And when 𝑥 is negative one, then the corresponding 𝑦-coordinate is negative a half.
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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.
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So in a, let’s try 𝑥 equals zero and see what the corresponding 𝑦-coordinate comes out to be.
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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.
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And anything to the power of zero is just one.
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So where 𝑥 is zero, 𝑦 is one.
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Well that we’ve fallen in the first hurdle there, we were hoping for 𝑦 to be negative one, so it’s not a.
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Let’s try b.
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And in equation b when 𝑥 is zero, we’ve got 𝑦 is equal to negative two to the power of negative zero.
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Well that’s the same as negative two to the power of zero, and two to the power of zero is just one.
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So this is negative one.
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Great!
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So it passes that test.
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Let’s try the next one then.
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When 𝑥 is one, we want 𝑦 to be equal to negative two.
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Well when 𝑥 is one, 𝑦 is negative of two to the power of negative one.
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Well two to the negative one is one over two, so that’s a half.
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So this is negative a half.
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But we wanted it to be negative two, so it fails the test.
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Let’s have a look at c then.
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Try 𝑥 equals zero.
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Well then 𝑦 is equal to the negative of half to the power of zero.
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Well half to the power of zero, anything to the power of zero is one, so that is negative of one.
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That’s negative one.
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That’s good.
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That’s what we wanted it to be, so we’re gonna have to try something else now.
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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.
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So let’s try when 𝑥 is equal to one.
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Well when 𝑥 is one, we’ve got 𝑦 is equal to the negative of a half to the power of one.
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Well a half to the power of one is just a half and the negative of that is negative a half.
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But when 𝑥 was equal to one, we wanted the 𝑦-coordinate to be negative two, so that is also wrong.
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Let’s hope it’s d then.
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Let’s try 𝑥 equals zero.
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When 𝑥 is zero, then 𝑦 is equal to negative two to the power of zero.
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The two to the power of zero is one, so this is negative one.
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Great!
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That looks like that works, so now let’s try 𝑥 equals one.
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And 𝑦 is equal to negative two to the power of one.
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Two to the power of one is just two, so the negative of two is negative two.
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And yep, that’s what we wanted.
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So now let’s just try 𝑥 is equal to negative one.
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And in that case, 𝑦 would be equal to the negative of two to the negative one.
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Well two to the negative one is one over two, so that’s a half.
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So that gives us negative a half, and that is in fact what we were looking for.
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So we validated all those three points.
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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.
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So this looks like the right answer.
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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.
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You think about it, two to the zero, two to the one, two to the two, two to the three, and so on.
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And then we’re taking the negative of all those 𝑦-coordinates, so we’re reflecting that in the 𝑥-axis.
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So it kind of matches on lots of fronts.
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So our answer is d.
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𝑦 is equal to negative two to the 𝑥.
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Now let’s consider this one, where would the graph of 𝑦 equals five to the power of 𝑥 intersect the 𝑦-axis?
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Well intersecting the 𝑦-axis means cutting the 𝑦-axis, and that means that it’s got an 𝑥-coordinate of zero.
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And when 𝑥 equals zero, then 𝑦 is equal to five to the power of 𝑥, so five to the power of zero.
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Five the power of zero is one.
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So our answer is that it would intersect the 𝑦-axis at zero, one.
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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?
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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.
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So let’s put that into our equation.
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So when 𝑥 is equal to zero, then 𝑦 would be equal to seven point one times six to the power of negative zero.
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Well negative zero is just the same as zero, so that’s six to the power of zero.
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Well anything to the power of zero, as long so it’s not zero, is one.
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So this becomes 𝑦 is equal to seven point one times one.
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Seven point one times one is clearly seven point one.
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So our answer is that it cuts the 𝑦-axis at zero, seven point one.
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The graph of the function 𝑦 equals 𝑏 to the power of 𝑥 passes through the point five, one.
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Find the value of 𝑏.
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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.
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So we can put those values for 𝑥 and 𝑦 into our equation and solve for 𝑏.
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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.
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Then 𝑦 is one, so that whole thing is gonna be equal to one.
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I’m gonna take the fifth root of both Sides so that I can work out what just 𝑏 is.
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So on the left-hand side, I’m gonna get the fifth root of one.
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And the fifth root of 𝑏 to the power of five is just 𝑏, which was obviously the point of taking fifth roots.
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Now the fifth root of one is just one, because one times itself, five times is one.
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So we found out that 𝑏 is equal to one.
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The graph of 𝑦 equals 𝑚 to the power of 𝑥 passes through the point three, twenty-seven.
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Find the value of 𝑚.
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And again this tells us that when 𝑥 is equal to three, then the corresponding 𝑦-coordinate would be equal to twenty-seven.
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So again we just need to plug those values of 𝑥 and 𝑦 into our equation and solve for 𝑚.
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So twenty-seven is equal to 𝑚 to the power of three.
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And this time I’m gonna take cube roots of both sides so that I can find out what just 𝑚 is.
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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.
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And the cube root of twenty-seven is three, so 𝑚 is equal to three.