Sunday, October 25, 2009

3.2 Reciprocal of a Quadratic Function- Period 3/4

Friday’s lesson was analyzing the reciprocal function of a quadratic function. With a linear function we took, for instance, y= x-7 and flipped it to be y= 1/ (x-7). The same goes for y= 2x+27 which in turn would be 1/(2x+27).

Because we are continuing with the theme of reciprocals, the same trend occurs with quadratic function, y= x ^2. The reciprocal of this function, just as it was with linear functions, is y= 1/ (x^2). Not too difficult, eh?

Now that we have our equation established, anything is possible. However rather than establishing rules to memorize, we’ll refer the now infamous comparison chart, which we have already done successfully in class for both linear and quadratic functions, to discuss the reasons for such similarities and differences. Why do the same concepts apply to both linear and quadratic functions and their reciprocals?

We’ll use the example y= 1/ x^2-4, shown above. I sincerely apologize for both graphs being in red.

  • Let’s first start with the zeros, or x-intercepts, of our function 1/x^2-4, which are -2 and 2. In these cases, the y-value of this coordinate is 0. When we apply this to the reciprocal, the denominator becomes 0, making the fraction 1/0, which doesn’t really work. So, two vertical asymptotes are created, in which the graph comes near however does not touch the x-values of 2 and -2.Next is the y-intercept, in which the x value is always 0 and for the quadratic function, the y-intercept is -4 while the reciprocal is, of course, -1/4.

  • Next are positive and negative intervals, which seem to follow the same trend for both linear and quadratic reciprocals. Why? Because if an x-value is positive, the reciprocal will of course be positive because the sign of the value is not changing. 2 will be 1/2 and -2 will be -1/2.

  • Lastly we’ll briefly review increasing and decreasing intervals. Simply put, as the x-values for the quadratic function increase, the reciprocals for these values decrease.


2-----> 1/2

3-----> 1/3

As the graph of the quadratic moves UP the reciprocal function moves DOWN. As the quadratic function INCREASES our reciprocal DECREASES.

1/2-----> 2

1/3-----> 3

As you can see, the same goes for fractions and negative x-values of our quadratic function and the x-values of the reciprocal quadratic function. As the x-values of the quadratic function DECREASE, the values of the reciprocal function INCREASE.

****For Ms. Burchat’s period 3/4 class:****

If you still don’t see the importance of conceptual understanding, recall her university horror story and be scared into caring about conceptual understanding! And another lifelong rule? Don’t let anyone call you slave.... Or esclave, servus, or esclavo. Or even sklave or abed for that matter.

And for some comic relief, check this out...

Don’t ask me how I found it. =)

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