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In this video, weβll learn how to sketch and identify the graphical transformations of exponential functions.
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An exponential function is one of the form π of π₯ equals π to the power of π₯.
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π is a positive real number not equal to one and the variable π₯ occurs as an exponent.
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These functions are hugely important within mathematics as they have all sorts of applications.
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We use them to model exponential growth and decay.
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For example, we might use an exponential function to model population growth or the amount of money in an investment account given specific compound interest requirements.
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Weβll begin by looking at the shape of such graphs.
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Which graph demonstrates exponential growth?
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First, letβs recall what we mean by an exponential function.
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Itβs a function of the form π of π₯ equals π to the power of π₯, where π is a positive real number not equal to one and in which the variable π₯ occurs as an exponent.
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Letβs look at what happens if we try to plot two types of this function.
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Weβll plot the function π of π₯ equals two to the power of π₯.
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In other words, one where π is greater than one and the function π of π₯ equals a half to the power of π₯.
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In this case, weβre looking at the behaviour where π is in the open interval from zero to one.
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Weβll use a table for each.
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When π₯ is negative two, π of π₯ is two to the power of negative two.
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Thatβs one over two squared, which is one over four or 0.25.
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When π₯ is negative one, π of π₯ is two to the power of negative one, which is a half or 0.5.
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In the same way, π of zero is one, π of one is two, π of two is four, and π of three is two cubed, which is eight.
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Similarly, for π of π₯, we get π of negative two to be four, π of negative one to be two, and so one.
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Letβs plot these on the same axes.
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When we plot the function π of π₯ and join it with a smooth curve, we see itβs increasing over its entire domain.
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In other words, itβs always sloping upwards.
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Whereas π of π₯ is decreasing; itβs always sloping downwards.
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We say, in fact, that a function of the form π of π₯ equals π to the power of π₯, where π is a real constant greater than one, represents exponential growth.
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Whereas when π is greater than zero and less than one, the function represents exponential decay.
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So which of our graphs demonstrates exponential growth?
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In other words, it looks a little bit like the function π of π₯.
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Well, we see that that function is π.
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Now, in fact, we can infer another property of these functions from the graphs we plotted.
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Notice how these parts of the lines seem to get closer and closer to the π₯-axis.
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Theyβll never actually reach the π₯-axis, though.
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And thatβs because the number gets fractionally smaller each time since weβre halving the value of the function.
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But it will never get to zero.
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We call this line, the π₯-axis or the line π¦ equals zero, a horizontal asymptote.
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We can therefore say the following.
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An exponential function is one of the form π of π₯ equals π to the power of π₯, where π is a positive real number not equal to one.
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If π is greater than one, the function models exponential growth.
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And if it is greater than zero and less than one, it models exponential decay.
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The π₯-axis, or the line π¦ equals zero, is a horizontal asymptote to such functions.
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Thereβs actually one further property that we can establish, so letβs look at an example.
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Determine the point at which the graph of the function π of π₯ equals six to the power of π₯ intersects the π¦-axis.
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We recall that the π¦-axis is the vertical line whose equation is π₯ equals zero.
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We can therefore find the point of intersection of the graph with the π¦-axis by letting π₯ equal to zero and solving for π¦.
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When we do, when we set π₯ equal to zero, we get π¦ equals six to the power of zero.
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But of course, we know that anything to the power of zero is one.
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This means that the graph intersects the π¦-axis at π¦ equals one.
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Now, thatβs of course when π₯ equals zero.
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So the coordinate of intersection is zero, one.
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In fact, if we take the general graph of a function π of π₯ equals π to the power of π₯, where π is a real constant greater than zero and not equal to one, we know that π of zero is π to the power of zero, which is also one.
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Remember, no matter the value of π, as long as itβs a real constant, π to the power of zero will always be one.
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We can therefore say that an exponential function of the form π of π₯ equals π to the power of π₯ intersects the π¦-axis at one or the point with coordinates zero, one.
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In our next example, weβll consider how we can identify the correct graph of exponential functions using the features weβve established and a bit of substitution.
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Which of the following graphs represents the equation π¦ equals three to the power of π₯?
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Our equation π¦ equals three to the power of π₯ represents an exponential equation.
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So letβs recall what we know about exponential functions.
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Firstly, we know that an exponential function of the form π of π₯ equals π to the power of π₯, where π is a positive real constant, passes through at zero, one.
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In other words, it passes through the π¦-axis at one.
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So letβs see if we can eliminate any of our graphs from our question.
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Graph B passes through at zero.
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Graph C doesnβt seem to intersect the π¦-axis at all.
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Graph E intersects at negative one.
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And that leaves us with Graph A and Graph D, which both intersect the π¦-axis at one.
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Thereβs two ways that we can check which one of our graphs is correct.
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We could pick a point and test this.
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For example, our first graph passes through the point with coordinates one, three.
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Letβs let π₯ be equal to one, since the π₯-coordinate is one, and see if the π¦-coordinate is indeed three.
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If π₯ is equal to one, π¦ is equal to three to the power of one, which is indeed three.
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And so we can infer that the graph of the equation π¦ equals three to the power of π₯ must pass through the point one, three.
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And so our graph is A.
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There is, however, another way we could have tested this.
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We know that if π is greater than one, our graph represents exponential growth.
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In other words, itβs always increasing.
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Whereas if π is between zero and one, it represents exponential decay; itβs always decreasing.
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We can see that the graph of D is decreasing over its entire domain.
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Itβs always sloping downwards.
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And so the value of π, the base if you will, needs to be between zero and one.
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So this could be π¦ equals one-third to the power of π₯, for example.
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The correct answer here then is A.
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Letβs have a look at another example.
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Which of the following graphs represents the equation π¦ equals a quarter to the power of π₯.
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Itβs useful to begin by spotting that this is an exponential equation.
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An exponential equation is one of the form π¦ equals π to the power of π₯, where π is a real positive constant not equal to one.
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Now, we know several things about the graphs of exponential equations.
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We know that their π¦-intercepts for a start are one.
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They pass through the point zero, one.
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And so we can instantly eliminate three of our graphs.
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We can eliminate A, B, and C.
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Graph A actually intersects at zero as does graph C, whereas graph B doesnβt appear to intersect the π¦-axis at all.
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Now, we also know something about the shape of these curves.
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If our value for π is greater than one, then weβre representing exponential growth.
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And the graph looks a little something like this.
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Notice that the π₯-axis represents a horizontal asymptote of our graph.
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It gets closer and closer but never quite touches it.
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Now, if π is greater than zero and less than one, we have exponential decay.
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Our graph is decreasing over its entire domain.
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The π₯-axis is still a horizontal asymptote to our graph, but this time it looks a little like this.
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So whatβs our value of π?
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Well, the equation is π¦ equals a quarter to the power of π₯.
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So π is equal to a quarter which is greater than zero and less than one.
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That tells us that our graph represents exponential decay.
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Itβs going to be decreasing over its entire domain.
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We can see that thatβs graph D.
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In our final example, weβll look at how to identify the graph of a more complicated exponential equation.
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Which of the following graphs represents the equation π¦ equals two times three to the power of π₯.
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Now, whilst it may not look like it, this is an example of an exponential equation.
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Itβs essentially a multiple of its general form π¦ equals π to the power of π₯, where π is a positive real constant not equal to one.
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This time, though, itβs of the form ππ to the power of π₯.
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Remember, according to the order of operations, we apply the exponent before multiplying.
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So this is three to the power of π₯ times two.
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And this means weβre going to need to recall what we know about the transformations of graphs.
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Well, for a graph of the function π¦ equals π of π₯, π¦ equals π of π₯ plus some constant π is a translation by zero π.
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It moves π units up.
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The graph of π¦ equals π of π₯ plus π is a translation by negative π zero.
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This time it moves π units to the left.
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Now, if we look at our equation, we see that we havenβt added a constant at all.
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So we recall the other rules we know.
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π¦ is equal to some constant π times π of π₯ is a vertical stretch or enlargement by a scale factor of π.
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Whereas π¦ equals π of ππ₯ is a horizontal stretch by scale factor one over π.
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Now going back to our equation, we have three to the power of π₯.
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And weβre timesing the entire function by two.
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And so weβre looking at a vertical stretch.
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In fact, we need to perform a vertical stretch of the function π¦ equals three to the power of π₯ by a scale factor of two.
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So what does the graph of π¦ equals three to the power of π₯ look like.
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Itβs an exponential function, and the base is greater than one.
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That means our function represents exponential growth.
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This means we can eliminate graphs A and B.
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They actually represent exponential decay, since theyβre decreasing; theyβre sloping downwards.
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So we need to choose from C, D, and E.
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And so we also recall that the function π¦ equals π to the power of π₯ passes through the π¦-axis at one.
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Our function π¦ equals three to the power of π₯ will do the same.
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Itβll pass through zero, one.
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But itβs been stretched vertically by a scale factor of two.
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This means our function π¦ equals two times three to the power of π₯ must pass through at zero, two.
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Out of C, D, and E, the only function that does so is E.
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C passes through at one and D passes through at three.
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And so the graph that represents the equation π¦ equals two times three to the power of π₯ is E.
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In this video, we learned that an exponential function is of the form π of π₯ equals π to the power of π₯, where π is a positive real number not equal to one.
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We saw that for values of π greater than one, our function models exponential growth; it slopes upwards.
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And that if zero is less than π, which is less than one, the function models exponential decay; it slopes downwards.
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Note that the reason we disregarded π equals one is if π is equal to one, the function gives a simple horizontal line, which is not exponential growth.
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Finally, we saw that these graphs pass through the π¦-axis at one and they have a horizontal asymptote given by the π₯-axis or the line π¦ equals zero.