Graph II : Modifying Graphs to Equations

This is the second part of my Graphs Blog. The first part showed how to convert Graphs to Equations. One important thing which I would like to add before I start the Blog, is that most Graphs do not not have a particular Equation, or equations for that matter, as in most graphs can't be limited to a few equations, but the equations can be found to plot the graphs very accurately.

There are many techniques involved in the art of graphing :

 1) Shifting - There are two types of shiftings, related to graphs. One is shifting the x axis, and the other shifting the y axis. 

    Given a function y = f(x), we have to find a function y = g(), such that the graph is shifted by k units to the right. Thus, we have x + k = . f(x) = g(), as the value of y will be constant, as only the x axis shifts. Therefore,   

                           f(x) = g(x + k)

                           g(x) = f(x - k)    [By replacing x with x - k]

   The y axis can be shifted in a similar manner.

                            g(y) = f(y - h)

   To cover it all, to shift a graph by k units to the right (- k to the left), or h units to the up (- h downward), convert all x to x - k, and all y to y - h.

The above is a Graph, and its Shifted Graph. (Credit : Desmos).

You can see that the initial graph(Black circle) has shifted its centre 2 units right and 3 units above.

2) Stretching - Stretching is exactly what it sounds like. Stretching the x axis of the graph is basically taking a graph and returning a graph multiplied by a factor, say k. Then, for every x in the original graph  y = f(x), the output graph returns an output function, y = g().

    = kx 

   f(x) = g(), as both y's are equal, as only the x is stretched.

   f(x) = g(kx)

   g(x) = f(x/k)

Similarly, to stretch the y axis by a factor of k, convert ofthe y's in the equation to y/k.

And, of course, both can be done together.

 

3) Superimposition - Superimposition is basically mixing more than one graphs, into one equation. This helps reducing the total number of equations. To explain this topic, look at this example : 

   f(x, y) = 

              = (x+y)(x-4y)

   f(x, y) = 0 means that either (x + y) or (x - 4y) are equal to 0.

   Thus, if we take the equation :

   

   It returns the graphs of (x + y = 0) and (x - 4y = 0) superimposed over each other.

4) Rotating Graphs - Rotating a graph by an angle θ, is basically rotating the axis by an angle θ. To rotate a graph, we just need to rotate each point separately, so if we can rotate a point (t, f(t)), we can rotate any point following that relation, i.e. the function to the new function, i.e. (x, y).

In the above diagram, O is the origin, and OX and OY the original x and y axis.  are the rotated axis. A is the point on the graph we need to rotate, and B is the rotated point. And we have , because that is how we drew it. Thus, OD = t, and BD = f(t), as they are the x and y coordinates on the new graph. Then, we can get :

Eliminating f(t), we get :

The black Curve is the original graph, the red one is the Graph rotated at an angle of π/6, and the blue line shows that the new graph is inclined at an angle of π/6.

5) Cutting Graphs - Cutting a graph is a pretty useful skill, because no one likes extras shapes outside the main figure required. It is like cutting the edges, kind of like, a finishing touch. To understand this, look at one example:

 

So, f(x) only exists in the real number line for x > 0.

And, that is how we removed one half on the graph. If we want to keep only the positive values of x, or y (swap x with y in the above function), in the graph of y = g(x), just multiply it by the above function.

The value won't change, as 1 is the Multiplicative Identity of any number.

If we multiply the function with f(-x), it will only show the negative part of the graph.

Over here, we can use another method, which I had explained earlier, shifting, to remove from certain parts, i.e, suppose you want the graph from a point b to infinity, you can shift the graph so that the b lies on the origin, and the shifting of the graph won't affect infinity at all.

If we want to shift f(x) 'b' units, we need to convert all x to x - b.

If we convert all x in f(-x) to x - b, we get the graph from negative infinity to b.

We can also figure out how to remove a specific section of the graph.

Suppose we want to cut out the part [-1, 1].

It is evident from the examples, and a bit of common sense, that we need to get function :

We also need a(x) < 0 ∀ x ∈ [-1, 1].

We can write that |x| - 1 ≤ 0.

If we want to keep only (-1, 1),

we need a(x) > 0 ∀ x ∈ [-1, 1].

If we multiply the a(x) for the previous function by -1, we get

1 - |x| ≥ 0

So, we can say that if we want to remove the graph from [-a, a], we need to multiply by :

And to keep only the part between (-a, a), multiply by :

Using this knowledge, we can effectively figure out a formula to effectively cut any section of the graph.

Suppose, you have to cut out the graph from [a, b], as in [a, b] are not included in the graph, we need to shift the origin, such that both the endpoints are equidistant, as we have in the formula. That point is the arithmetic mean of a and b, i.e. . So, if we shift the graph by that many units, we get the new origin. Now, we need to find it's distance from the endpoints, i.e,

So, effectively, if we want to remove that part, [a, b] from the graph, we get the new formula to multiply to the equation of the graph :

To keep (a, b), we need to multiply the graph by :

And, that's how you do it.

We can use this information to plot many interesting shapes in the graph.

We can also plot alphabets on the graph and write names on the graph.

The second pic is just an extension of the first equation.

These equations will give :

It is clear that there is Cutting is every equation. In some equations, Shifting of the graph is also apparent. You can also see that the 'S' is Schrodinger is formed using rotating the sine wave curve. 

Try to keep the number of equations as little as possible, like, I kept all the parts, which are between 

y = -3π/2 and y = 3π/2 in Equation 1.

Similarly, we can draw any name, and many shapes on the graph.

This one equation give :

And that's how you do it...