Graphing Reciprocal Functions
Reciprocal of a Linear Function y = x
We will start with a simple linear function y=x.
Every y-value is the same as the x-value, so we can
fill in the rest of the table pretty quickly.
Lets complete a table of values for
this simple function.
x
y
-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
This is a pretty simple function, since the y value is
just the same as the x value.
Points That do not change when you find their reciprocal.
What is the reciprocal of 1? 1
So the reciprocal of 1 is 1.
1
1
I will plot the values for the reciprocal function with green
on the same grid as y=x. (see below)
1
What is the reciprocal of -1?  1  1 So the reciprocal of -1 is -1.
So now we have two points drawn on the graph of the reciprocal of y=x.
1
1
This reciprocal function is known as yy 

x
x
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.
Let’s find some more points
When x is –100, y= -100 on the line y=x.
The When
reciprocal
is
–1/100,
which
is
x4,isy=
– 1/2,
y=the-1/2
ony=x.
thevery
linesmall
y=x. (too small to graph)
When
x
is
–
-4
on
line
When
x
is
–
1/4,
y=
-1/4
on
the
line
y=x.
We observe
that as xisapproaches
very graphed
negativebelow.
value (negative infinity),
The reciprocal
– 2 which
isa now
The
reciprocal
isfunction
–1/4,
which
is now
graphed
below.
The
reciprocal
is
–
4
which
is
now
graphed
below.
the
reciprocal
approaches
a
small
negative
value (it actually approaches zero)
When x is – 2, y= -2 on the line y=x.
On the
graph, this
means
that is
y=0
is agraphed
horizontal
asymptote (drawn in green).
The
reciprocal
is –1/2,
which
now
below.
The reciprocal graph approaches this line, but never touches it.
The reciprocal function HUGS the asymptote.
f ( x) 
-100
-100

-4
-4
1
100
1

4
-2
-2

Absolute value of y is small. (less than 1)
We already have the points (1,1) and (-1,-1) figured out.
Absolute value of the reciprocal y-value is small (less than 1)
1
x
y
x

1
2

1
4

1
2

1
4
Absolute value of the reciprocal y-value0is big. 0
(greater than 1)
1
2
-2
-4
undefined
Ifatwe
at thethe
absolute
value of(blue
the y values
(that is ignore
the negatives)
weto
saw
Another way to lookLet
thislook
is when
base function
increasing
(going
up from left
right),
us
make
some general
observations line)
aboutiswhat
we did here.
that
if
the
original
y-value
was
less
than
one
(examples:
¼,
½)
then
the
reciprocals
So
for
reciprocal
functions,
big
original
y-values
become
small
then the reciprocal function
(red
is decreasing
(going
down(that
fromisleft
to right),
If we look
at curve)
the
absolute
ofthe
the
y y=x.
values
ignore
the negatives) we saw
When
x ispoints
0, yvalue
=one
0 on(examples:
line
y-values
greater
than
4, 2) Remember
weofare
If were
wey-values
join
the
with
a smooth
curve,
we will big
getthat
half
theignoring
graph ofthe
reciprocal
and
small
original
y-values
become
reciprocal
y-values.
that if the original
y-value was
larger
than
one (examples:
100, 4, 2) then the reciprocals
The
reciprocal
is
1/0
which
is
undefined.
Division
negatives
this
discussion.
thein
function
y =back
1/x . into
Thisplay,
is drawn
in redthat
on original
the graphnegative
above. y-values
To bring
the
negatives
we notice
y-values were by
less
than
one
(examples:
¼, ½)
Remember that we are ignoring the
zero
is not
defined.
There 1/100,
is a vertical
asymptote
result
reciprocal y-values. So we just make sure that we graph negative y-values
at xdiscussion.
= 0. (the y-axis). The reciprocal graph approaches
Absolute value of y is big.(greaternegatives
than
1) ininnegative
this
this line, but never touches it. The reciprocal function HUGS
the asymptote. The asymptote is drawn in blue on the graph above.
The second half of y = 1/x
Original y-values greater than 1
We will now look at positive x-values.
Reciprocal y –values greater than 1
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 less 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)
x
y=1/x
¼
4
½
2
1
1
2
½
4
¼
If we draw a smooth curve through these points, we get the
other half of y = 1/x. (Drawn in red above)
Here is the table of values
Note that the graph of the base function y = x is increasing.
for some positive x-values.
(goes up from left to right) and the reciprocal function is
decreasing (goes down from left to right)
Summary
The table below summarizes what we have learned.
f(x)
1
f ( x)
1
1
-1
-1
0
asymptote
Negative
Negative
Positive
Positive
Big
Small
Small
Big
Increasing
Decreasing
Decreasing
Increasing
Quick and Easy
We did it 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 yvalues – 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.
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