These are some records and observations made during EMT 668 Assignment 1
Activity 2

Activity 2.

Make up linear function f(x) and g(x). Explore, with different pairs of f(x) and g(x) the graphs for

i. h(x) = f(x) + g(x)
ii. h(x) = f(x) . g(x)
iii. h(x) = f(x) / g(x)
iv. h(x) = f(g(x))

Summarize and illustrate.

I will approach this from a purely exploratory perspective - that is I will attempt to explore the properties of teh four cases using Algebra Expressor as a tool of exploration. The observations I will make will be based on the information gleaned from the graphs drawn by Algebra Expressor rather than on any algebraic manipulations. I justify this as follows: I believe that I am capable of the algebraic manipulations. However, I am trying to make a case for the use of the computer as a tool of investigation and would like to demonstrate its power by using it as such!

In this activity I will explore all four cases at a time. For each case I will report the values of the variables a,b,c,d as in:

f(x) = ax + b
g(x) = cx + d

Let us begin with the case where f(x) and g(x) have the same co-efficients of x but different constants :


fig 1: a = c = 1, b = 2, c = -2

Let us change the co-efficients of x to be opposite in sign and see what happens:


fig 2: a = 1, b = 2, c = -1, d = -2

This time let us explore what happens when the coefficients of x are different both in magnitude and opposite in sign:


fig 3: a = 2, b = 2, c = -1, d = -2

Next we explore the case where the co-efficient of x is opposite in sign but similar in magnitude and the two constants are the same:


fig 4: a = 1, b = 2, c = -1, d = 2

Finally let us consider the case where the coefficients are both opposite in sign and magnitude while the constants are the same:


fig 5: a = 2, b = 2, c = -1, d = 2

Without considering greater differences in the actual numbers used we are in a position to consider soem trends and make some conjectures. To begin with let us summarise what we have seen:

SUM OF FUNCTIONS: h(x) = f(x) + g(x)

h(x) will be a straight line, both the y intercept and gradient are the sum of the y intercepts and gradients of f(x) and g(x). A horizontal line is possible if the gradients of f(x) and g(x) are equal and opposite in sign. No vertical lines are possible

PRODUCT OF FUNCTIONS: h(x) = f(x) + g(x)

As can be expected (linear x linear = 2nd degree equation) the result is a parabola. The shape of the parabola is effected by:

"arms-up parabola" when gradients of f(x) and g(x) have the same sign,
"arms-down parabola" when gradients of f(x) and g(x) are opposite in sign,
x-intercepts of the parabola x1 and x2 are the x intecepts of f(x) and g(x),
the gradients of the linear functions will determine the number of cuts of the original linear functions with the parabola - THIS IS THE SUBJECT OF ACTIVITY THREE

QUOTIENT OF FUNCTION: h(x) = f(x) / g(x)

h(x) is an interesting function. There are two different shapes adopted by h(x)

(i) h(x) is a straight line parallel to the xaxis, and

(ii) h(x) is a rectangular hyperbola with asymptotes paralel to the x and y axis.

based on the cases explored so far it is too early to conjecture

(i) when we will get a straight line rather than a hyperbola, and

(ii) when the hyperbola will be in the "1st and 3rd quadrants" vs "2nd and 4th quadrants"

some further exploration will reveal answers to these questions.........

FUNCTION OF A FUNCTION: h(x) = f ( g (x) )

This is not as interesting as the previous cases - the result is a straight line (linear function) - which is quite easy to understand based on the structure of h(x). What remains is to prepare a conjecture on how the coefficients of x in f(x) and g(x) determine the gradient of h(x).

To go to activity 3 (WRITE UP 1) click here.

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