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In this video I'll talk a little bit about how a function can be viewed as a graph.
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So far, we've been viewing a function the following ways. Remember a function is an action.
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It takes an input x, it does something to it, here is the box f that does something to x and then we get an output, which we call f(x).
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So to go back to our original example, the tripling function, we could describe that in words.
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Take the input and, eh, triple it.
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So that describes what the function does. And equivalently, we can describe the function with an equation.
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So, here it is, f(x) equals 3x. This is whatever the input is, multiplied by three and that's your output.
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So let's think how to view this as a graph.
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First I'll make a table of some pairs of numbers. The input here and the output, there.
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So our input,... maybe our input is -2, our output would be -6. Because we take the input, multiply it by 3, -2 times 3 is -6.
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An input of 0 gives an output of 0, because 3 times 0 is 0.
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An input of 2 gives an output of positive 6, because 2 times 3 is six.
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And, let's do one more. An input of 4 gives an output of 12, because 4 times 3 is 12.
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So I can take these pairs of numbers and plot them on an x-y plane. So let me sketch what that would look like and then I'll show you a nicer printout.
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So first I draw these axis.
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So, the y or the vertical will be f(x), the output. And the inputs are down here, on the horizontal axis.
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And then let me put on a scale.
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There are my axis. And now I can plot some points.
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Maybe I'll plot these points in a different color so they stand out.
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Let's do -2 and -6. So -2, that's down here on the x-axis. I would go over 2 and then down 6. So that point might be there.
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(0,0), that's the origin, right here in the middle. This is the point where x is zero and y is zero.
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(2,6), let's plot that; over 2, up 6. It'a around there.
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And then (4,12), over 4, up 12, there it is. So those are my four points.
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So, I'm just taking these numbers and plotting them and this function turns out to be a line. It's a linear function.
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So I can just connect the dots. I do that in purple, and that would be a graph of the function.
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So this purple line is the function. It carries the same information as this word version or the equation version. I could say I want a function that triples the input,
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I could describe the function by saying f(x) = 3x, or I could describe the function just by showing you this graph.
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And the point is that all contain the same information. Let's look at a nicer version of this graph.
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Here's one I had a computer draw for me.
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We can see it looks pretty similar to the graph I drew before.
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And again this graph contains the same information as this formula. So I could use this graph to figure out an input - output pair.
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For example, suppose I'm interested in f(10). If we have 10 and the function f acts, what happens? Well, I would read that off from the graph.
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Here is my input value 10. I go up to the graph that represents the function and then I would read on the vertical axis that this value is about 30.
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Of course, we wouldn't necessarily need to use this graph because we have the formula, but the point is if we didn't have the formula we could still figure out values of functions
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by reading off the graph.
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Let's do another graphical example. Here is a function, here is the graph.
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And for variety I've called this h(x) instead of f or g. And we don't have a formula and we don't have a verbal description of this function.
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So, in order to figure out input and output values, we 'll have to use the graph.
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So, let's see how that would go.
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Suppose we're interested in h(4). So, if 4 is the input, what is the output? Alright, so the input value, that's down here, that's 4.
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And then I would say what's the h(x) if x is 4? That's what this curve tells me. The height of the curve is the value of h(4) and I would read that off and say "oh, it's around 10".
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So h(4) is around 10. And I could put a squiggly equal sign there to indicate that it's an approximation. Maybe the curve doesn't exactly go through 10. We can't tell but that's okay.
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Let's do a few more points.
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h(2). So if 2 is the input what is the function? Let's see. So here is my x value of 2. h(2) is the height of the curve.
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And I would look here at the curve and then read over here and I'd say "well, that's probably around six". Again, it's just approximate, but that's okay.
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Let's do one more. Maybe h(-2). So now -2 is my input. What's the function value? That's the height of the curve here and I'll look and I'll say "that's about 2.5".
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Not exact, and it's an approximation, but that's okay.
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So to summarize, we can represent a function with words or an equation or with a graph, and all three are equivalent.
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Specifying the graph of a function is the same as giving a formula. You can figure everything out from the graph that you could from the formula.
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Often though, working with a graph is much easier and we'll do that a bunch in both this unit and next.