Make up linear functions 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. h(x)=f(x)+g(x)
Based on a simple computation, it is easy to see that h(x) has a slope that is the sum of f(x)'s and g(x)'s and a y-intercept that is the sum of f(x)'s and g(x)'s. So let's go onto something more intersting such as
ii. h(x)=f(x)g(x)
Using the same kind of computation as in (i) gives a quadratic equation. By changing to vertex form we see how the linear equations influence the quadratic. We can read off the location of the vertex and whether the parabola will be concave up/down, right from the vertex form. Using the quadratic formula we could, if we had the energy, predict the roots of the quadratic.
iii. h(x)=f(x)/g(x)
The graph for h(x) will generally look something like this. The horizontal intercept, horizontal asymptote, and vertical asymptote are straightfoward to compute. Using the first and second derivative we can also get a lot of information. Note that if g(x) is constant and f(x) is not then h(x) is another linear function.Clearly, the problem implies g(x) is nonzero for otherwise h(x) will not make any sense. If f(x)=0 then h(x) is the x-axis (h(x)=f(x)). Finally, if f(x) is constant then h(x) looks alot like 1/x except for possibly a translation and/or a dilation.
iv. h(x)=f(g(x))
In this case, h(x) will be another linear function. A straightfoward computation tells us what linear function h(x) will be.