Quarter Wit, Quarter Wisdom: Bagging the Graphs

Quarter Wit, Quarter WisdomOne thing that I would like to suggest to increase speed in Arithmetic is Multiplication Tables. Much to my dad’s chagrin, I am still a little lost when confronted with 16×7 or 17×8 or 18 × 7 (I know the last two are 126 and 136 but in what order, I am not sure) but rest I can pretty much manage. And many a times, while solving little toughies, I have blessed my dad for his incessant reproach regarding tables in days yonder.

Now, the one thing that I would like to suggest to increase speed in Coordinate Geometry and Algebra is learning how to draw graphs. Know how to draw a line from its equation in under ten seconds and you shall solve the related question in under a minute. For now, take my word for it and go ahead.

This is what the xy coordinate axis looks like:

If we were to draw the line x = 2 this is what it would look like:

On this line, at every point, x coordinate is 2. y coordinate varies from point to point.

Similarly, y = -1 is as shown:

Taking a cue from above, can you tell me, how you would draw y = 0? Of course it is the X-axis. Where is y = 0? At every point of the X axis!

And then the equation of y axis must be … yes, x = 0.

But usually the kind of lines we need to draw, look something like this:

Any given line on the xy plane can be uniquely described using two characteristics – the line’s slope and a point through which the line passes.

Slope of a line:

The slope of the line is just a measure of how tilted it is. As x increases, if y increases much more, the line becomes more titled. Look at the blue line below. When x increased by 1 (from -1 to 0), y increased by 2 (from 0 to 2) so the line has a slope of 2. The green line below has a slope of 1. When x increases by 1 unit, y also increases by 1 unit. On the other hand, the red line has a very small slope. When x increases by 1 unit, y increases by very little. So what about the orange line? There, when x increases, y decreases! Of course the slope is negative there. The purple line has a slope of -1. When x increases by 1 unit, y decreases by 1 unit. So we see that when you go from left to right (à), if the line is going up, its slope is positive; if it is going down, its slope is negative.

A Point on the line:

The second characteristic that defines a line is a point through which it passes. This could be expressed in many ways: y intercept, x intercept or (x, y)

When I say y intercept of a line is 4, it just means that it passes through (0, 4) i.e. it cuts the y axis at point 4. When I say the x intercept of a line is -2, it just means the line passes through (-2, 0) i.e. it cuts the x axis at point -2. Or I could simply say that the line passes through (1, 6). If I have any one of these and the slope, I can draw a unique line. Let’s try it.

Given that the slope of a line is -2 and its y intercept is 4, how will you draw the line?

First of all, since slope is -2, the line will look something like this

Now, since it cuts the y axis at 4, the line can be drawn like this:

So do the questions tell you that slope is -2 and y intercept is 4? At GMAC, they don’t really like you that much! What they generally give is the equation of a line, say 2x + y – 4 = 0. You will re-arrange this equation to get y = -2x + 4 (Remember y = mx + b where m is the slope and b is the y intercept?). You get -2 as the slope and 4 as the y intercept.

Any equation can be put in this format.

3x + 4y -6 = 0. Re-arrange to get y = -3x/4 + 6/4. Slope = -3/4 and y intercept = 3/2.

Soon, we will discuss another quick method of drawing a line given its equation.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep in Detroit, Michigan, and regularly participates in content development projects such as this blog!

6 Responses

  1. Himanshu says:

    Hi Karishma,
    Excellent Article. However I need an explanation- In the last graph, what are the coordinates of the line at X -axis? Is it (2,0)

    • Karishma says:

      Hey Himanshu,

      You are right. The slope of the line is -2. This means that if y co-ordinate decreases by 4, the x co-ordinate will increase (because of negative slope) by 2 (because change in y is double the change in x)
      (0, 4) lies on the line. So if we make the y coordinate 0 (i.e. decrease it by 4), the x co-ordinate will increase by 2. So (2, 0) will lie on the line.

      Hope it all makes sense now.

      • Shadab Khan says:

        Or simply we can directly put x=0 in the equation y = -2x +4
        which will give y = 2. that means it is intersecting x axis at (2,0).
        However Karishma’s approach is better that this. :)

        • Karishma says:

          Yes Shadab, you are right. You can simply put y = 0 to get the x co-ordinate and know where the line intersects the x axis. However, the focus of this post is on ‘why’ i.e. why will the line intersect the x axis at the point (2, 0). It should make sense to you that since it has slope -2 and it passes through (0, 4), it will pass through the point (2, 0). It focuses less on the process of arriving at the answer and more on why the process is what it is. Once you understand the ‘why’ of these concepts well, you can solve many questions quickly by just some little manipulations.
          Hope it all makes sense.

  2. Shivam says:

    Hi Karishma,

    Can you please draw the number line for |1-x|<2 ? i have got lil confused !!

    Thanks :)

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