Examples and limitations of graphing multivariable functions.
What we're building to
- Graphing a function with a two-dimensional input and a one-dimensional output requires plotting points in three-dimensional space.
- This ends up looking like a surface in three-dimensions, where the height of the surface above the -plane indicates the value of the function at each point.
Reviewing graphs of single-variable functions
Graphs are, by far, the most familiar way to visualize functions for most students. Before generalizing to multivariable functions, let's quickly review how graphs work for single-variable functions.
Suppose our function looks like this:
To plot a single input, like , we first compute :
Then we mark the point on the -plane. In this case, that means marking .
When we do this for all possible inputs , not just , we see what all the points of the form look like.
Unless is some exotic or sporadic function that gives wildly different values as changes slightly, the result will be a smooth-looking curve.
Adding one more dimension
So what can we do for functions with a two-dimensional input and a one-dimensional output? Perhaps something like this:
Associating inputs with outputs requires three numbers—two for the inputs and one for the output.
To represent these associations using a graph, we plot points in three dimensions.
- The association is plotted with the point .
- The association is plotted with the point .
- In general, the goal is to represent all points of the form for some pair of numbers and .
The resulting graph is shown below. The video shows this graph rotating, which hopefully will help you get a feel for the three-dimensional nature of it. You can also see the -plane—which is now the input space—below the graph.
This means for any given point on the plane, the vertical distance between that point and the graph indicates the value of . That vertical direction is usually referred to as the -direction, and the third axis which is perpendicular to the -plane is called the -axis.
As long as the value of changes continuously as and change values, which is almost always the case in functions we deal with in practice, the graph ends up looking like some sort of surface.
Example 1: The bell curve
Gaussian bell curve
Let's analyze what's going on with this function. First, let's look inside the exponent of and think about the value .
Question: How can you interpret the value ?
When the point is far from the origin, the function will look like , which is nearly zero. This means the distance between the graph and the -plane at those points will be tiny. When and , on the other hand, , which is what gives us the bulge in the middle.
Reflection Question: The graph above has rotational symmetry, in the sense that it will look the same if we rotate it in any way about the -axis. Why is this?
Example 2: Waves
One way to get an intuition for the function —and multivariable functions in general—is to see what happens when one of the inputs is held constant.
For example, what happens when we fix the value of to be ? Usually, we are plotting all the points that look like this:
By holding constant at , we are limiting our view to points that look like this:
There is a very nice way to interpret this geometrically:
The points in space where , which is to say all the points of the form , make up a plane. Why? Imagine slicing the graph with this plane. The points where the plane and the graph intersect—drawn in red above—are the points on our graph where .
So why would this be helpful for understanding the graph?
We have basically turned the multivariable function into a single variable function:
In fact, the curve we get from slicing the three-dimensional graph at has the same shape as the two-dimensional graph of .
In this way, you can understand the three-dimensional graph of a multivariable function one slice at a time by holding one variable constant and looking at the resulting two-dimensional graph.
Example 3: One input, two outputs
You can also graph a function with a one-dimensional input and a two-dimensional output—although, for whatever reason, this is not commonly done.
In this case, only runs freely, while the and values on the graph are both dependent on .
If we rotate the image so we can look squarely at the -plane, the graph looks like . Another way to say this is that when we project the graph onto the -plane, it gives the graph of .
Projection of onto the plane
Similarly, rotating the image so that we're looking squarely at the -plane makes the image look like the graph of .
Projection of the onto the -plane
In other words, this function is a way to combine the two functions and into one, and its graph captures the information of both in one image.
As soon as you try to apply this process to functions with higher-dimensional inputs or outputs, you'll run out of dimensions which you can comfortably visualize.
For instance, consider the function with a two-dimensional input and a two-dimensional output. Graphing it would require four dimensions of space! This is because we would need to plot all points of the form .
In practice, when people think about graphs of higher dimensional functions, like , they usually start by considering graphs of some simpler function with a two-dimensional input and one-dimensional output like . This is a sort of conceptual prototype.
Such a prototype can help make sense out of certain operations, and can give you a feel for what starts to happen as your input space becomes multidimensional. At the end of the day, the actual computations are performed purely symbolically for the higher dimensional function.
Want to join the conversation?
- can anyone show me how can i plot these 3d graphs on matlab or mathematica or maxima or scilab or anyother app? :((8 votes)
- Found the link below really helpful
- can you make everything a little clearer? I am having trouble understanding(3 votes)
- In order to discriminate what each value (or set of values), when pumped into the function, will return, you must have one dimension per variable.
Take this analogy: you have a bag of apples, and a bag of oranges. You want to express how many you gain per second, at any given second, and make a model of it. So you section the floor into "blocks" and label each block with a time. Then you take however many apples, and however many bananas that you gained and place a apple or banana in that section, respectively.
We can't display how many apples AND bananas we gained by only placing apples in the marked off section: we need both apples AND bananas so people can discriminate between them, right? Same thing with a graph. We can't display two values with only one dimension: we need (at least) one dimension per value.
All is great if we only have 2 or 3 values in total, but what happens when we get to 4? Or 5? Or 100? Can you imagine a 100 dimension grid? I can't.
I hope I helped. Please tell me if I completely missed what you were struggling with.(7 votes)
- In the second paragraph under the header 'Limitations', f(x,y) = (x^2,y^2) is said to be a function of two inputs and one output, while in the third paragraph, f(x,y,) = (x^2 + y^2) is said to be a function of two inputs and one output.
I don't understand the how f(x,y,) can represent either one output, or two. Also, I don't understand why different notation is used between (x^2,y^2) and (x^2 + y^2) if they are both two inputs. Sorry if the question is unclear or doesn't make sense!(2 votes)
- Hi there, do not get confused, output is given in the form of f(x,y) this means that we are getting two outputs by putting the value of x and y in to the function, so this function produces two values one is x and y for each given or certain value of x and y. Eg: F(x,y) = sin(x) , tg (y) here I have independent values x and y this is so clear, to my function I am producing one sin(x) output for x values, and one tg (y) output for any y. Pay attention; If I write F(x,y) = sin(x).tg(y) in this case my function returns only one value for each given x and y. ( dot and semicolon)(4 votes)
- How do I learn benchmark angles? I can't seem to understand it. Please help me Khan Academy Guirdians(3 votes)
- In example 3, what would the image look like if we looked at it from the yz plain?(2 votes)
- In the 𝑦𝑧-plane, the points plotted are (𝑥², sin(𝑥)),
which means that the curve would look like the functions
𝑓₁(𝑥) = sin(√𝑥) and 𝑓₂(𝑥) = −sin(√𝑥).(3 votes)
- Shouldn’t the function with two outputs and one input, when looked at so only the x and z plane are visible, look like a sinusoidal function whose period decreases as distance from the origin increases?(2 votes)
- In example 3, why would f(x) = sin(x) outputs in xz plane ? Why not in xy, like f(x) = x² ?(2 votes)
- because in xy plane only parabola is getting formed do a little math on this you will get it(1 vote)
- What is the difference between a surface and curve? Thanks(1 vote)
- The curver is a curved line in the multi-dimentional space, while the surface typically has more dimensions. You've already seen 2-dimentional surfaces above. Also for example there exists an opinion that our universe is a 3-dimentional surface wrapping 4-dimentional sphere of space-time. I hope I gave you a clue :)(2 votes)
- "Such a prototype can help make sense out of certain operations, and can give you a feel for what starts to happen as your input space becomes multidimensional."
Please elaborate. What operations are we talking about here?(1 vote)
- In Example 3 one input three outputs how is the height of the spiral growing in height (seems to be growing) when it is a sin function moving in waves(1 vote)
- I also first thought it looked that way, but it is just an "optical illusion". The square term for y = f(x) = x^2 grows very large fast and this can sort of give the impression, that z is also growing, when in reality, it is only y. It is just the viewing angle combined with having a single constant thickness line.(1 vote)