GRAPH SKETCHING Methods20060810
Boring Basics:
Two axes at right angles: the x-axis is horizontal [“X is a Cross”] and the y-axis is vertical. The numbering on each axis must be consistent [meaning that the spacing between numbers must be even], but they don’t have to be to the same scale. Most of the time we use the little squares in our books as units (of one). The axes meet where both x and y are zero. So all along the x-axis y is zero; all along the y-axis, x is zero. A point on a graph is written as an (x,y) pair. These are conventions, like spelling.
Method One (better for Year Eight). Example: x + 2y = 10
Where is x zero (for this equation)? 0 + 2y = 10 means 2y = 10 means y = 5 (which you might do in your head)
so mark y=5 on the y-axis (remember, x=0 on the y-axis). So you have marked (0,5)
Where is y zero? x + 0 = 10 means x = 10. So mark x=10 on the x-axis ...... the point (10,0)
What sort of line is this? Is it a straight line? Yes it is: more later
The join the dots.... Join (0,5) and (10,0) with a straight line, going through the points (draw the line as far as your axes go.
A sketch does not need to be to scale: it is allowable to draw two axes, mark five and ten on the relevant axes and join them. It is even allowable to draw the axes and the line and mark where they meet as five and ten. You must label the axes x and y. This method can be very quick.
You try:
x + 4y = 8 x + 4y = 8 2x – 3y = 126x – y = 12
3x - 2y = 6 –x + 4y = 12 2x + 3y = +6 4x – 3y = 12
Method Two (as taught by everyone else and which doesn’t belong in Year Eight; I’ve given you the straight theory and then written the method the way it works in your head)
The equation of a straight line is of the form y = mx + c. the letter m represents the gradient – how far up the page you go for each one that x changes. The letter c represents the y-intercept, where x is zero. so the method goes like this:–
Where is x zero (for x + 2y = 10)? where y = 5; so mark y=5 on the y-axis; you have marked (0,5)
What happens when x changes by one? Well, you just worked out where x = 0, so what next easiest? x = 1? So when x = 1 for our equation, 1 + 2y = 10; 2y = 9; y = 4.5. Mark (1, 4.5) on your graph.
How did the graph change? It went down by a half unit; the gradient is -0.5 (say “minus a half”)
Do I need some more points? Usually the answer to this is ‘yes’, so go across one more, work it out or check, and draw (2,4), (3, 3.5) and maybe (4,3).
Do they make a straight line? Well, if they don’t, you goofed: this method starts from the assumption that you recognised the equation as a straight line (which is one argument why Method One is better). If they don’t, fix it. It is sensible to check that (for this equation) (10,0) is on your line.
What a ‘sketch’ IS:
A sketch has labelled axes;
it has the shape of the line (straight or curved);
it shows the ‘interesting’ places, particularly the places where the line meets the axes (intercepts) and where lines meet (intersections). For curves, interesting places include where the line is vertical or horizontal, where the line changes curvature, where the line disappears to...
I distinguish between sketch, draw and plot: If you carefully drew it to scale, you overdid things and drew the graph.If you were ever so careful and put in lots of points, you plotted the graph. Sketching is the quick and dirty method that lets you see what is going on with a problem.
Maths is a tool for doing things, not an end in itself: you want a sketch to help with a problem, not to become the problem!
Adding to this in 2013: Sketching is a skill used throughout school maths. In Middle School (KS4 especially), one is more likely to be forced by a question to draw, having produced a table of points to plot. The reasoning is quite understandable: examiners need to test the number skills covered by creating a table (can you put numbers into a formula?), then test ability to choose sensible scales to fit a page, to plot points correctly, connect them appropriately and, often, use this graph to generate some sort of answer, preferably to a problem not likely to be solved exactly with the maths available to the expected students. My 2012/3 students seem to appreciate that their sketches are drawn quickly enough to avoid many problems with the subsequent attempt to draw.
