Here are some more examples of Reciprocal functions.

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Example 1 of a Reciprocal Function

•Original Y-values of zero result in vertical asymptotes for the reciprocal

•Original negative y-values result in reciprocals that are negative (the reciprocal will not cross the x-axis)

•Where the original was decreasing, the reciprocal is increasing (original big y-values result in reciprocal small y-values; original small y-values result in reciprocal big y-values – don’t forget we talked about the absolute value)

•Where the original was decreasing, the reciprocal is increasing (original big y-values result in reciprocal small y-values; original small y-values result in reciprocal big y-values)

•Original Y-values of 1 and –1 don’t change

•Original positive y-values result in reciprocals that are positive (the reciprocal will not cross the x-axis)

2

1)(

xxf

First draw y= - x +2, which is a straight line with slope -1 and y-int 2.

We did it!

The graph hugs the asymptotes

Here is a better picture of the function

Example 2 of a Reciprocal Function

•Original Y-values of zero result in vertical asymptotes for the reciprocal

•Where the original was increasing, the reciprocal is decreasing (original big y-values result in reciprocal small y-values; original small y-values result in reciprocal big y-values

•Where the original was decreasing, the reciprocal is increasing (original big y-values result in reciprocal small y-values; original small y-values result in reciprocal big y-values)

•Original Y-values of 1 and –1 don’t change

•Original positive y-values result in reciprocals that are positive (the reciprocal will not cross the x-axis)

2

1)(x

xf

First draw y=-x2, which is a parabola with vertex at (0,0).

We did it!It kind of looks likea skinny volcano.

The graph hugs the asymptotes

Here is a more accurate graph.

Example 3 of a Reciprocal Function

•Original Y-values of zero result in vertical asymptotes for the reciprocal.But for this one the original function does not haveany y-values of zero, so there are no vertical asymptotes.

•Where the original was increasing, the reciprocal is decreasing (original big y-values result in reciprocal small y-values; original small y-values result in reciprocal big y-values

•Where the original was decreasing, the reciprocal is increasing (original big y-values result in reciprocal small y-values; original small y-values result in reciprocal big y-values)

•Original Y-values of 1 and –1 don’t changeBut for this one there are no y-values of 1 or –1. So we take the reciprocal of the vertex which would be ½.

•Original positive y-values result in reciprocals that are positive (the reciprocal will not cross the x-axis)

2

1)(

2 x

xf

First draw y=-x2, which is a parabola with vertex at (0,0).

We did it!It kind of looks likea speed bump.

The graph hugs the asymptotes

Here is a more accurate graph

Another Example of a Reciprocal Function

•Original Y-values of zero result in vertical asymptotes for the reciprocal

•Where the original was increasing, the reciprocal is decreasing (original big y-values result in reciprocal small y-values; original small y-values result in reciprocal big y-values

•Where the original was decreasing, the reciprocal is increasing (original big y-values result in reciprocal small y-values; original small y-values result in reciprocal big y-values)

•Original Y-values of 1 and –1 don’t change

•Original positive y-values result in reciprocals that are positive (the reciprocal will not cross the x-axis)

2)2(

1)(

x

xf

First draw y=-x2, which is a parabola with vertex at (0,0).Then shift the parabola two units to the right.

We did it!It kind of looks likea skinny volcano.But shifted two units right.

The graph hugs the asymptotes

Here is a more accurate graph.

Another Example of a Reciprocal Function

•Original Y-values of zero result in vertical asymptotes for the reciprocal

•Where the original was increasing, the reciprocal is decreasing (original big y-values result in reciprocal small y-values; original small y-values result in reciprocal big y-values

•Where the original was decreasing, the reciprocal is increasing (original big y-values result in reciprocal small y-values; original small y-values result in reciprocal big y-values)

•Original Y-values of 1 and –1 don’t change

•Original positive y-values result in reciprocals that are positive (the reciprocal will not cross the x-axis)

4

1)(

2 X

xf

First draw y=-x2, which is a parabola with vertex at (0,0).Then shift the parabola four units down.

We did it!It kind of looks likea pig – a pig? See the next slide to see what IMean.

The graph hugs the asymptotes

•Original negative y-values result in reciprocals that are negative (the reciprocal will not cross the x-axis)

It looks like a pig!Or a dog or whatever you can imagine – just come up with something to help you remember.