Blog 4

Hello everyone.

My name is Mohammed Alqahtani and I'll be explaining Curve Sketching. 

Curve sketching is an art as well as a science. This art can take the understanding and visualization of algebraic concepts to another level. As they say, a picture is worth a thousand words. Let me start with an example of this. Let us say, I tell you that a quadratic equation is something where the dependent variable varies as the square of the independent variable. Someone new to math and functions would almost certainly be confused to hear this description. Now, if instead of providing the verbose definition of a quadratic equation, I simply draw the following parabola (Figure 1), it is much easier to visualize, understand and appreciate.

Figure 1: Parabolic graph representing the quadratic equation y = x²

Having seen the instant understanding and visualization that is imparted by curve sketching, let us move on to some basic methods of sketching a curve. The first and foremost is by means of what I call an “x-y table”. This table is nothing but a tabulated version of entries of x and corresponding calculated entries of y. Once you have generated a sufficiently long list, you simply need to create the x and y axes on a piece of paper and plot the points from table. Then join those points to get the graph. Let us consider the graph of y = sinx for an example. In order to plot it, we would first create the x-y table (see Table 1 below). In the table, we would plot x (independent variable, angle) and calculate y for each x by taking the sine of angle x (Figure 2).

x	y = sinx
0	0
30	0.5
60	0.866
90	1
120	0.866
150	0.5
180	0

Table 1: x-y table for y = sinx

Figure 2: Graph for y = sinx

Now that we have seen how curve sketching can be done for a given function, let us consider how to modify an existing curve. This is where curve sketching becomes a bit of an art and not just a science. We would briefly discuss some ways of modifying an existing graph.

We just now saw the graph of y = x². Let’s say we now need to plot the graph for y = x²+1. There are two ways we can approach this. First way is to prepare an x-y table for this new function and then plot it accordingly (as we did for sinx graph just now). The other way (a more artistic way) is to modify the graph for y = x² by moving it UP by 1 unit along y-axis as shown in Figure 3.

Figure 1: Parabolic graph representing the quadratic equation y = x²+1

Likewise, other rules exist for translating the graph UP or DOWN and RIGHT or LEFT, as summarized in Table 2.

Function	Transformation
y = f(x) + a	Move graph of y = f(x) UP by a units
y = f(x) - a	Move graph of y = f(x) DOWN by a units
y  = f(x+a)	Move graph of y = f(x) LEFT by a units
y = f(x-a)	Move graph of y = f(x) RIGHT by a units

Table 2: Rules governing translations of a function to create another function

Now that we have covered these concepts, we can continue enhancing our understanding of these ideas. There are similar ideas, when it comes to generating the graph of y = f(-x), which can be obtained by taking the reflection of the graph for y = f(x) about y-axis. Likewise, -y = f(x) is obtained by taking the reflection of y = f(x) about x-axis. We can talk about this subject for hours and continue to explore the various ways of curve sketching and modification. However, the best way to learn this is to practise these concepts on your own. Just like a picture is worth a thousand words; doing something yourself is worth much more than reading or listening about it. You gain so much more by practising the concepts on your own. You will be surprised how much you can learn on your own.