A major problem today concerns educating the next generation of engineers, mathematicians and researchers. Too many of our nation's students end up neither comprehending nor liking math courses. More intensive drilling of material as currently practiced may be both ineffective and undesirable. In fact, this rigorous drilling may turn more young students away from mathematics and the sciences. Why has this situation developed when mathematics is so interesting? There are situations and mathematical principles that will enable graphs of functions to be easily produced. This paper will provide and discuss principles that can be applied in graphing a large class of functions. The graphs of polynomials and special basic functions formed by functional composition acting on polynomials will be provided as examples of visual thinking. Engineering students who are encouraged to develop the skills of visual thinking in mathematics may find these skills beneficial in their analytical engineering studies. A student could find pleasure and confidence in discovering the ability to gain insight into the graphical behavior of a large class of functions. This same student may become more open to studying other aspects of polynomials and other functions. These techniques can provide a quick check of computer-generated graphs or be employed when a computer is unavailable or inconvenient. If we desire to recruit more students into the analytical and other sciences, we need to discover better, easier and more pleasurable ways to present conventional math concepts before attempting to accelerate curricula by moving advanced differential concepts into the lower grades.
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