Graphing Reciprocal Functions

Reciprocal of a Linear Function y = x

We will start with a simple linear function y=x.

Lets complete a table of values forthis simple function.

x y

This is a pretty simple function, since the y value isjust the same as the x value.

-100 -100

-2 -2

-1 -1

-1/2 -1/2

-1/3 -1/3

0 0

1/3 1/3

1/2 1/2

1 1

2 2

100 100

Every y-value is the same as the x-value, so we canfill in the rest of the table pretty quickly.

Points That do not change when you find their reciprocal.

What is the reciprocal of 1? 11

1 So the reciprocal of 1 is 1.

What is the reciprocal of -1? 11

1

So the reciprocal of -1 is -1.

I will plot the values for the reciprocal function with green on the same grid as y=x. (see below)

A y-value of 1on the function y=x is also on the reciprocal function because its value does not change.

A y-value of –1on the function y=x is also on the reciprocal function because its value does not change.

So now we have two points drawn on the graph of the reciprocal of y=x. This reciprocal function is known as

xy

1x

y1

Let’s find some more points

x yx

xf1

)(

-100 -100

-4-4 4

1

-2-22

1

2

1

2

1 -2

4

1

4

1 -4

0 0

We already have the points (1,1) and (-1,-1) figured out.

When x is –100, y= -100 on the line y=x.The reciprocal is –1/100, which is very small (too small to graph)We observe that as x approaches a very negative value (negative infinity),the reciprocal function approaches a small negative value (it actually approaches zero)On the graph, this means that y=0 is a horizontal asymptote (drawn in green). The reciprocal graph approaches this line, but never touches it. The reciprocal function HUGS the asymptote.

100

1

When x is – 4, y= -4 on the line y=x.The reciprocal is –1/4, which is now graphed below.When x is – 2, y= -2 on the line y=x.The reciprocal is –1/2, which is now graphed below.

When x is – 1/4, y= -1/4 on the line y=x.The reciprocal is – 4 which is now graphed below.

When x is 0, y = 0 on the line y=x.The reciprocal is 1/0 which is undefined. Divisionby zero is not defined. There is a vertical asymptote at x = 0. (the y-axis). The reciprocal graph approachesthis line, but never touches it. The reciprocal function HUGSthe asymptote. The asymptote is drawn in blue on the graph above.

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If we join the points with a smooth curve, we will get half of the graph of the function y = 1/x . This is drawn in red on the graph above.

Let us make some general observations about what we did here.If we look at the absolute value of the y values (that is ignore the negatives) we saw that if the original y-value was larger than one (examples: 100, 4, 2) then the reciprocals y-values were less than one (examples: 1/100, ¼, ½) Remember that we are ignoring the negatives in this discussion. Absolute value of y is big.(greater than 1)

Absolute value of the reciprocal y-value is small (less than 1)

Another way to look at this is when the base function (blue line) is increasing (going up from left to right),then the reciprocal function (red curve) is decreasing (going down from left to right),

Absolute value of y is small. (less than 1)

Absolute value of the reciprocal y-value is big.(greater than 1)

When x is – 1/2, y= -1/2 on the line y=x.The reciprocal is – 2 which is now graphed below.

So for reciprocal functions, big original y-values become smallreciprocal y-values and small original y-values become big reciprocal y-values. To bring the negatives back into play, we notice that original negative y-values result in negative reciprocal y-values. So we just make sure that we graph negative y-values.

If we look at the absolute value of the y values (that is ignore the negatives) we saw that if the original y-value was less than one (examples: ¼, ½) then the reciprocals y-values were greater than one (examples: 4, 2) Remember that we are ignoring the negatives in this discussion.

The second half of y = 1/x

We will now look at positive x-values.

For positive y-values, the reciprocals will also be positive.Remember that original y-values that are big (greater than 1) will result in small reciprocal function y-values (less than 1)

Original y-values greater than 1

Reciprocal y-values less than 1

For positive y-values, the reciprocals will also be positive.Remember that original y-values that are small (less than 1) will result in big reciprocal y-values (greater than 1)

original y-values less than 1

Reciprocal y –values greater than 1

If we draw a smooth curve through these points, we get theother half of y = 1/x. (Drawn in red above)

Note that the graph of the base function y = x is increasing. (goes up from left to right) and the reciprocal function is decreasing (goes down from left to right)

x y=1/x

¼ 4

½ 2

1 1

2 ½

4 ¼

Here is the table of values for some positive x-values.

SummaryThe table below summarizes what we have learned.

f(x)

1 1

-1 -1

0 asymptote

Negative Negative

Positive Positive

Big Small

Small Big

Increasing Decreasing

Decreasing Increasing

)(

1

xf

Quick and Easy •Original Y-values of 1 and –1 don’t change•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 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 – don’t forget we talked about the absolute value)

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

•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 – don’t forget we talked about the absolute value)

This one does not apply in this case because the original function was always increasing. Some lines might be always decreasing and parabolas will have some parts that are increasing and others parts that are decreasing.

We did it quick and easy!

Conclusion

• We can graph a reciprocal function y=1/f(x) by first graphing y=f(x) and then note the following

• Original y-values of 1 and –1 don’t change• Original y-values of 0 result in vertical asymptotes• Where the original function was increasing the

reciprocal function is decreasing (and vice versa)• The reciprocal function does not cross the x-axes