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The Humongous Book of Algebra Problems by W. Michael Kelley

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Chapter 16
GRAPHING FUNCTIONS
Graphing linear equationsand therefore graphing linear functions—is a
straightforward task. There are numerous techniques by which the graphs
can be constructed (described in Chapter 5), but the behavior of a linear
graph is predictable. Other functions, however, present a challenge. Cubic,
absolute value, and square root functions, to name a few, have distinctly dif-
ferent graphs.
Rather than design algorithms by which each classication of function
should be graphed, it is more useful to use one of two techniques to plot
function graphs: completing a table of values (that is, the brute force tech-
nique) to plot an extremely accurate graph or applying function transfor-
mations to create a relatively accurate sketch of the graph quickly.
Graphing functions is tricky. It wouldnt be if all functions were lines,
because let’s face it, linear graphs are the easiest to create. All you need is
one point and the slope of the line and youre in business! However, most functions
arent linear, so you need a few more graphing strategies.
There are basically two ways to draw a function graph. One is to plug in a lot
of x-values into a function f(x) to see what numbers you get as a result. Plug in
enough xs and you can connect the dots to make a graph. The downside? A lot
of calculations and a big time investment.
There is a faster way to graph that involves memorizing a few basic function
graphs and then using transformations to stretch, squish, move, and ip them
to make new graphs. For example, after you memorize the simple graph of y = x
3
,
you can graph things like f(x) = x
3
+ 6, g(x) = (x – 7)
3
, and h(x) = –2x
3
in the blink
of an eye.
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