Friday, October 23, 2009

Characteristics of Quadratic Reciprocal Functions

In the last two lessons we have talked about reciprocal functions. To review a reciprocal is the flip of any regular function of a function in the form 1/f(x) instead of just f(x). Some easy examples of reciprocals are things like the reciprocal of 2 is 1/2, and 3 is 1/3. A reciprocal function is y=x+2, and y=1/(x+2).

Quadratic reciprocals
A quadratic reciprocal is the same as an ordinary reciprocal function ex. y=x^2 +2, y=1/(x^2+2). It also follows the same rules as a regular degree 1 reciprocal function:
  • The zeros (or x-intercepts) of a quadratic will be the asymptotes of the reciprocal function. Just like in a degree 1 reciprocal function except for the fact that for some quadratics there are 2 zeros, instead of just 1. Reason: You cannot have 1/0 it does not work. So on the original graph, f(x), the only values that made the equation =0 were the x-intercepts. This means the x intercepts cannot happen on a reciprocal graph or else it would equal 1/0.
  • When the regular quadratic has positive y values the reciprocal function will also have positive y values, same for negative. Reason: 2 is +, 1/2 (its reciprocal) is also positive. -2 is -, -1/2 its reciprocal is also negative.
  • Where the original graph is increasing (from left to right) the reciprocal function will be decreasing and the opposite. Reason: 2 the reciprocal is 1/2, 3 the reciprocal is 1/3, 4 is 1/4, and 5 is 1/5. This shows that as the original gets bigger the reciprocal gets smaller showing why as one graph increases the other decreases.
  • Finally all the points on the original graph have reciprocal points on the second graph.
These 4 common rules for all reciprocal functions once mastered will allow you to always graph a reciprocal function. I was going to make 2 graphs to show but I have no idea how??? Some one tell me on Monday and I can make it better but until then this is all I've got.

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