
UNIT 3 : QUADRATIC FUNCTIONS & EQUATIONS
LESSON
8: RECIPROCAL FUNCTIONS
Graphing
functions and their reciprocals:
Example 1:
x 


4 
 6 
1/6 
2 
 4 
 Ό 
1 
 3 
1/3 
0 
2 
½ 
1 
1 
1 
1.5 
0.5 
2 
1.75 
0.25 
4 
2 
0 
1/0 = undefined 
2.25 
0.25 
4 
2.5 
0.5 
2 
3 
1 
1 
4 
2 
½ 
5 
3 
1/3 
Please note the following from the graph and table:
·
The
graph of y = f(x) = x 2 (blue) is a line and its reciprocal (red) has 2 branches separated by the line x = 2 (dashed). It is called a
hyperbola.
·
Where
f(x) has a zero (x intercept), the reciprocal has
an asymptote (x = 2 dashed)
·
The
behaviour near the asymptote is interesting;
as x approaches 2 from the right (x = 3, 2.5, 2.25, 2.1
), the
reciprocal (red) gets very large in the positive
direction
·
As x
approaches 2 from the left (x = 1, 1.5, 1.75, 1.9
), the reciprocal (red) gets very large in the negative direction.
·
As x
takes on larger positive values, the reciprocal takes on smaller positive
values approaching zero from above.
·
As x takes on larger negative values (1,2, 4, 10,
), the reciprocal takes
on smaller negative values approaching zero again from below.
·
Where
f(x) is positive, the reciprocal is positive;
where f(x) is negative, the reciprocal is negative.
·
Where
f(x) = 1, the reciprocal equals 1;
where f(x) = 1 , the reciprocal equals 1. [Points (1, 1), (3, 1)]
Example 2:
Solution:
First
find the zeros or x intercepts of y = x + 3. Let y = 0 and solve for x.
0
= x + 3 and
x =  3
Hence x =  3 is
the x intercept of f(x)
and the vertical asymptote of the reciprocal.
Find all
points on the line where y=1 or y=1.
[(2, 1) and (4, 1)]
Now make
a table of values as in above example including values on either side of the
asymptote x = 3.
x 


11 
8 
1/8 
6 
3 
1/3 
5 
2 
½ 
4 
1 
1 
3.5 
0.5 
2 
3.25 
0.25 
4 
3.1 
0.1 
10 
3 
0 
Undefined 
2.9 
0.1 
10 
2.75 
0.25 
4 
2.5 
0.5 
2 
2 
1 
1 
1 
2 
½ 
0 
3 
1/3 
1 
4 
Ό 
8 
11 
1/11 
x =3
Example 3:
Solution:
First
find the zeros or x intercepts of the parabola f(x). Let y = 0 and solve for x.
Now make
a table of values if needed as below including points near the asymptotes x =
2 and x = 4. See red graph below.
x 


4 
12 
1/12 
3 
7 
1/7 
2.5 
3.25 
0.31 
2.1 
0.61 
1.64 
2.01 
0.0601 
16.64 
2 
0 
Undef. 
1.99 
0.0599 
16.69 
1.9 
0.59 
1.69 
1 
9 
1/9 
3.9 
0.59 
1.69 
4 
0 
Undef 
4.1 
0.61 
1.64 
5 
7 
1/7 
6 
12 
1/12 
Note values of x taken near asymptotes:
x =  2.1, 2.01,  1.9, 1.99,
3.9, 4.1
Please note the following from the graph and table:
·
The
graph of y = f(x) = x^{2} 2x
 8 (blue) is a parabola with vertex at (1, 9) and zeros
2, 4. Its reciprocal (red) has 3 branches separated by the
lines
x =  2 and x
= 4(dashed).
·
Where
f(x) has a zero (x intercept), the reciprocal has
an asymptote (x =  2 and x = 4 dashed)
·
The
behaviour near the asymptotes is interesting;
as x approaches  2 from the right (x = 1.9, 1.99 in table), the
reciprocal (red) gets very large in the negative
direction (goes down); as x approaches
 2 from the left (x =  2.1, 2.01 in table), the reciprocal (red) gets very large in the positive direction (goes up). Similar behaviour occurs near the other
asymptote x = 4.
·
As x
takes on larger positive values (x = 5, 6 in table), the reciprocal takes on
smaller values approaching zero from above.
As x takes on larger negative values ( 3,  4 in table), the reciprocal
takes on smaller values approaching zero again from above.
·
Where
f(x) is positive, the reciprocal is positive; where f(x) is negative, the reciprocal is
negative.
·
Where
f(x) = 1, the reciprocal equals 1;
where f(x) = 1, the reciprocal equals 1. [Points (+/ 4.2, 1) and (+/ 3.8, 1)].
Example 4:
Given the
graph of f(x) below, sketch the graph of the reciprocal.
Solution:
The zero of f(x) is x = 2. This is the vertical asymptote of the reciprocal.
Take
points on the graph of f(x) and use the mapping below.
(6, 4) ΰ (6, Ό )
(4, 2) ΰ (4, ½ )
(3, 1) ΰ (3, 1)
(2, 0) ΰ V. Asymp.
(1, 1) ΰ (1, 1)
(2, 4) ΰ (2, Ό )
(4, 6) ΰ (4, 1/6)
Example 5:
Given
the graph of f(x) below, sketch the graph of the reciprocal.
Solution:
The zeros of f(x) are x = 4 and x = 4. These are the vertical asymptotes of
the reciprocal. Take points on the graph of f(x) and use
the mapping below.