How are the graphs of the reciprocal functions related to their corresponding original functions? What happens to the graphs of
the reciprocal functions as x approaches the zeros of the original functions? Describe how you would teach friends with different learning styles (visual-spatial, aural-auditory, verbal-linguistic, physical-bodily-kinesthetic, logical-mathematical, social-interpersonal, and solitary-intrapersonal) how to graph the reciprocal functions.
We have in general that when a function has a high value, its reciprocal has a high value and vice-versa. That is the correlation between the function. When the function goes close to zero, it all depends on the sign. If the graph approaches 0 from positive values (for example sinx for small positive x), then we get that the reciprocal function is approaching infinity, namely high values of y. If this happens with negative values, we get that the y-values of the function approach minus infinity, namely they have very low y values. 1/sinx has such a point around x=0; for positive x it has very high values and for negative x it has very low values. It is breaking down at x=0 and it is not continuous.
Now, regarding how to teach it. The visual way is easy; one has to just find a simulation that makes the emphasis as the x value changes and shows us also what happens if we have a coefficient 7sinx and 1/(7sinx). If they have a more verbal approach to learning, it would make sense to focus on the inverse relationship between a function and its reciprocal... and also put emphasis on the importance of the sign of the function when the function is near 0. Logical mathematical approach: try to make calculations for large values of x and small values of x, introduce the concept of a limit of a function (Where its values tend to) or a function being continuous (smooth).
