In this video I'll talk a little bit about how a function can be viewed as a graph.
So far, we've been viewing a function the following ways. Remember a function is an action.
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).
So to go back to our original example, the tripling function, we could describe that in words.
Take the input and, eh, triple it.
So that describes what the function does. And equivalently, we can describe the function with an equation.
So, here it is, f(x) equals 3x. This is whatever the input is, multiplied by three and that's your output.
So let's think how to view this as a graph.
First I'll make a table of some pairs of numbers. The input here and the output, there.
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.
An input of 0 gives an output of 0, because 3 times 0 is 0.
An input of 2 gives an output of positive 6, because 2 times 3 is six.
And, let's do one more. An input of 4 gives an output of 12, because 4 times 3 is 12.
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.
So first I draw these axis.
So, the y or the vertical will be f(x), the output. And the inputs are down here, on the horizontal axis.
And then let me put on a scale.
There are my axis. And now I can plot some points.
Maybe I'll plot these points in a different color so they stand out.
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.
(0,0), that's the origin, right here in the middle. This is the point where x is zero and y is zero.
(2,6), let's plot that; over 2, up 6. It'a around there.
And then (4,12), over 4, up 12, there it is. So those are my four points.
So, I'm just taking these numbers and plotting them and this function turns out to be a line. It's a linear function.
So I can just connect the dots. I do that in purple, and that would be a graph of the function.
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,
I could describe the function by saying f(x) = 3x, or I could describe the function just by showing you this graph.
And the point is that all contain the same information. Let's look at a nicer version of this graph.
Here's one I had a computer draw for me.
We can see it looks pretty similar to the graph I drew before.
And again this graph contains the same information as this formula. So I could use this graph to figure out an input - output pair.
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.
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.
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
by reading off the graph.
Let's do another graphical example. Here is a function, here is the graph.
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.
So, in order to figure out input and output values, we 'll have to use the graph.
So, let's see how that would go.
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.
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".
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.
Let's do a few more points.
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.
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.
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".
Not exact, and it's an approximation, but that's okay.
So to summarize, we can represent a function with words or an equation or with a graph, and all three are equivalent.
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.
Often though, working with a graph is much easier and we'll do that a bunch in both this unit and next.