Monday 21 May 2018

ALG 15

We continue with Cartesian graphs for another week - sorry, bad planning! (But worth it, hopefully.)
In this week's tasks we foster a feel for graphs by 'adding' or 'subtracting' them. We begin with a visual approach, by focussing on the 'vertical' distance between two graphs - especially when this is zero, ie where the graphs intersect.
We go on to relate the visual to the symbolic by expressing the graphs symbolically and adding or subtracting the relations.
MONDAY: Here we have two linear functions, f(x) and g(x), though we don't know precisely what they are since the Cartesian axes have not been numbered. However, we can, for example, say that the graph of the new function y = f(x) – g(x) cuts the y axis at the same point as f(x) cuts the axis [Why?], and cuts the x axis directly below the point of intersection of the given graphs [Why?].
It is interesting to note that f(x) is not as steep as g(x). What does this tell us about the slope of y = f(x) – g(x)?
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TUESDAY: Here we are given some feedback on Monday's task and have the opportunity to consolidate our earlier ideas.
We are are also given one of the functions in symbolic form which allows us to determine the scale of the axes and hence to represent all the other functions symbolically. We can thus link the symbolic with our earlier visual/numerical/analytic approach.
We can use the symbolic representations in various ways. For example, knowing that f(x) = x + 10 and that g(x) = 2x, we can state that y = f(x) – g(x) = x + 10 – 2x which simplifies to y = 10 – x. We can then use this symbolisation to check whether our original sketch (ie the purple line) is correct, or, we could derive the symbolisation from the sketch, given that we now can see that the purple line goes through points with coordinates (0, 10), (10, 0) and (20, -10)
In the case of the function y = f(x) + g(x), we can determine that its line will have a gradient of 3, on the basis that we are adding lines with gradients of 1 and 2, or on the basis that its equation will contain the terms x and 2x, whose sum is 3x.
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WEDNESDAY: Here we start with a very familiar straight line graph (of the function y = x) and 'perturb' it by adding a second, 'wilder' function. Interestingly, though, the effect of this second function is quite localised....
Note: I came across this lovely idea in Abraham Arcavi's 1994 Symbol Sense paper in For the Learning of Mathematics, 14, 3.
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THURSDAY: Here we can check whether our sketch for Wednesday's task was on the right lines by comparing it to the red curve: this expresses the fact that the term 1/(x – 4) has a very large effect on the value of y when x is very close to 4, but its effect rapidly diminishes as we move away from x = 4. We then get a chance to build on this by sketching the graph of a closely related function.
The green curve below shows the graph for the new function. Since 2x – 8 = 2(x – 4), the term 1/(2x – 8) has only half the effect of the term 1/(x – 4).
FRIDAY: Here we subtract a linear function from a quadratic function.
The effect turns out to be surprisingly simple: just a translation of the curve (see below). Can we use algebra to find the translation (and to show the shape hasn't changed)?

Interestingly, if one subtracts a similar linear function from a cubic, as below (blue – orange), the resulting cubic curve (grey) is not congruent, or even similar, to the original cubic curve.

Next week: The Arithmagons are coming: Captain Scarlet's bitter foes (or was that the Mysterons?).