In this way we can systematically view the effects of changing the operation as well as the effects of negative functions.
These graphs show us that the sum of two linear functions is a linear function. Notice too, that the slopes and y-intercepts of h(x) are simply the sum of the slopes and y-intercepts respectively of f(x) and g(x). (The reader may notice that this is similar to vector addition and vaguely similar to multiplication of complex numbers, where the angles are added!) Thus when just f(x) or g(x) is negative, h(x) is more of a "mix" of the two, while it is a more obvious addition of the two in the first and fourth frames. One can also see that the negative of a linear function is a reflection in the line through the x-intercept of that function. In other words in figures 1 and 2 the red lines [f(x) and -f(x)] are a reflection about the line x=-1.
We see clearly from the above figures that the product of two linear
functions is a quadratic function. It is also readily apparent either by
viewing the graphs or doing the algebra, that when f(x) and g(x) are both
negative we obtain the same graph as we when they are both positive since,
-1(-1)=1. Similarly, regardless of which function is negative [f(x) or g(x)],
h(x) is the same. Notice also that in the bottom figures h(x) is the negative
of h(x) in the top figures. Thus the parabolas are "upside down",
or reflected about the x-axis.
So we see that division of two linear functions results in a hyperbola. The equations corresponding to the top figures are in fact the same. Likewise the equations corresponding to the bottom figures are equal. Furthermore, the bottom equation is simply the negative of the top, resulting in a reflection about the x-axis.
Again from these pictures, it is clear that composition of two linear
functions produces a linear function. Since we are composing f of g of x,
we notice that taking the negative of f(x) results directly in the negative
of h(x) [compare the top figures as well as the bottom figures]. On the
other hand, the negative of g(x) totally changes h(x) [Compare the left
figures, top and bottom. Now compare the right figures, top and bottom.].