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Reciprocal Functions

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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

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)]

 

Text Box: Strategies for plotting linear reciprocal graphs:
·	Find the zero (x – int) of the original graph.  This is the asymptote of the reciprocal.
·	Make a table of values like above including points near the asymptote. 
·	Use the 1/x key on your calculator to get the reciprocal values in the third column of the table.
·	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.
 

 

 

 

 

 

 

 

 

 


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

-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) =  x2 – 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)].

 

 

Text Box: Strategies for plotting reciprocal quadratic graphs:
·	Find the zeros (x – int’s) of the original graph.  These are the asymptotes of the reciprocal
·	Find the vertex and its reciprocal.  This gives a point between the asymptotes.
·	Use the 1/x key on your calculator to get the reciprocal values in the third column of the table.
·	Where f(x) is positive, the reciprocal is positive;  where f(x) is negative, the reciprocal is negative.
·	 Where f has large positive values, the reciprocal will have small positive values near zero
·	 Where f has large negative values, the reciprocal will have small negative values near zero
·	Where f(x) = 1, the reciprocal equals 1;  where f(x) = –1, the reciprocal equals –1.
·	Make a table of values like above including points near the asymptotes. 
·	Use the mapping  (x, y)  ΰ  (x, 1/y)  to determine the points on the reciprocal graph.
 

 

 

 

 

 

 

 

 

 

 

 

 

 


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.

 

           

 

           

 

 

 

 

 

 

 

 

 

 

           

 

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