L E S S O N 9 - 2 A N D L E S S O N 9 - 3

THE RECIPROCAL FUNCTION FAMILY AND RATIONAL

FUNCTIONS AND THEIR GRAPHS

ASSIGNMENT 2/12/15

• Section 9-2 (p506)

• 2, 6, 16, 22, 24, 28, 30, 32

• section 9-3 (p513)

• 1 – 18

• Functions that model inverse variations belong to a family whose parent is the reciprocal function

• 𝒇 𝒙 =𝟏

𝒙, 𝐰𝐡𝐞𝐫𝐞 𝒙 ≠ 𝟎.

• Transformations

• 𝒇 𝒙 =𝒂

𝒙+ 𝒌 𝒎𝒐𝒗𝒆𝒔 𝒖𝒑

• 𝒇 𝒙 =𝒂

𝒙− 𝒌 𝒎𝒐𝒗𝒆𝒔 𝒅𝒐𝒘𝒏

• 𝒇 𝒙 =𝒂

𝒙−𝒉 𝒎𝒐𝒗𝒆𝒔 𝒓𝒊𝒈𝒉𝒕

• 𝒇 𝒙 =𝒂

𝒙+𝒉 𝒎𝒐𝒗𝒆𝒔 𝒍𝒆𝒇𝒕

• a is the stretch (if 𝒂 > 𝟏) 𝒐𝒓 𝒔𝒉𝒓𝒊𝒏𝒌 𝒊𝒇 𝟎 < 𝒂 < 𝟏

• 𝒂 < 𝟎 is a reflection in the x-axis

GRAPHING AN INVERSE VARIATION

• Sketch a graph of 𝒚 =𝟔

𝒙, 𝒙 ≠ 𝟎

• What are the asymptotes?

• Each part of the graph is

called a branch.

GRAPHING AN INVERSE VARIATION

• Sketch a graph of 𝒚 =𝟑

𝒙, 𝒙 ≠ 𝟎

• What are the asymptotes?

• Each part of the graph is

called a branch.

GRAPHING RECIPROCAL FUNCTIONS

• Draw the graph of 𝒚 =−𝟒

𝒙

• Describe the

transformations

GRAPHING RECIPROCAL FUNCTIONS

• Draw the graph of 𝒚 =−𝟐

𝒙

• Describe the

transformations

• A musical pitch is determined by the frequency of vibration of the sound waves reaching the ear.

• The greater the frequency, the higher is the pitch.

• Frequency is measured in vibrations per second, or hertz

(Hz).

• The pitch (y) produced by a panpipe varies inversely with

the length (x) of the pipe.

• Write the function:

• Find the length of the pipe that produces a pitch of 277 Hz.

• Pitches of 247 Hz and 370 Hz. Find the length of pipes that

will produce each pitch.

• The asymptotes of this equation are y=0 and x=0. Explain why this makes sense in terms of the panpipe.

• Desmos Graphing Calculator

xy

564

GRAPHING TRANSLATIONS OF RECIPROCAL FUNCTIONS

• Graph on desmos

• 𝒚 =𝟏

𝒙, 𝐲 =

𝟏

𝒙−𝟏, 𝒚 =

𝟏

𝒙+𝟐

• What are the vertical and horizontal asymptotes for each graph?

• How do the vertical asymptotes relate to the denominators equaling zero?

• Now graph 𝒚 =𝟏

𝒙, 𝒚 =

𝟏

𝒙+ 𝟏, 𝒂𝒏𝒅 𝒚 =

𝟏

𝒙− 𝟐

• What are the vertical and horizontal asymptotes for each graph?

GRAPHING A TRANSLATION

• Sketch the graph of 𝒚 =𝟏

𝒙−𝟐− 𝟑

GRAPHING A TRANSLATION

• Sketch the graph of 𝒚 = −𝟏

𝒙+𝟕− 𝟑

GRAPHING A TRANSLATION

• Sketch the graph of 𝒚 = −𝟕

𝒙−𝟖−4

WRITING THE EQUATION OF A TRANSFORMATION

• Write an equation for the translation of 𝒚 =𝟓

𝒙 that

has asymptotes at 𝒙 = −𝟐 𝒂𝒏𝒅 𝒚 = 𝟑.

WRITING THE EQUATION OF A TRANSFORMATION

• Write an equation for the translation of 𝒚 = −𝟏

𝒙

that has asymptotes at 𝒙 = 𝟒 𝒂𝒏𝒅 𝒚 = 𝟓.

WRITING THE EQUATION OF A TRANSFORMATION

• Write an equation for the translation of 𝒚 =𝟏𝟑

𝒙 that

has asymptotes at 𝒙 = 𝟓 𝒂𝒏𝒅 𝒚 = −𝟖.

SECTION 9-3

• Objective:

• Students will identify properties of rational

functions

• The graph is continuous because it has no

jumps, breaks, or holes in it.

• It can be drawn with a pencil that never

leaves the paper.

• There is no real value of x that makes the

denominator 0

• The graph is not continuous since there is a

break

• X can not be 2 and -2

• The graph is not continuous.

• X can not be -1

A point of discontinuity is either a hole or a vertical

asymptote.

FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY.

12

)4(32

xx

xy

For each rational function, find any points of discontinuity.

The function is undefined at values of x for which x2 – x – 12 = 0.

x2 – x – 12 = 0 Set the denominator equal to zero.

(x – 4)(x + 3) = 0 Solve by factoring or using the

Quadratic Formula.

x – 4 = 0 or x + 3 = 0 Zero-Product Property

There are points of discontinuity at x = 4 and x = –3.

a. y = 3

x2 – x –12

x = 4 or x = –3 Solve for x.

FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY.

12

12

xx

y

FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY.

1

12

x

xy

FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY.

3

12

2

x

xy

FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY.

16

)4(2

x

xy

FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY.

25

)5(22

x

xxy

FOR EACH RATIONAL FUNCTION, FIND ANY POINTS OF DISCONTINUITY.

82

)4)(1(2

xx

xxy

Describe the any points of discontinuity ( vertical asymptotes and

holes for the graph of each rational function).

Since –1 and –5 are the zeros of the denominator and neither is a zero

of the numerator, x = –1 and x = –5 are vertical asymptotes.

a. y = x – 7

(x + 1)(x + 5)

–3 is a zero of both the numerator and the denominator.

The graph of this function is the same as the graph y = x,

except it has a hole at x = –3.

b. y = (x + 3)x

x + 3

c. y = (x – 6)(x + 9)

(x + 9)(x + 9)(x – 6)

6 is a zero of both the numerator and the denominator.

The graph of the function is the same as the graph y =

which has a vertical asymptote at x = –9, except it has a hole at x = 6.

1

(x + 9) ,

DESCRIBE THE ANY POINTS OF DISCONTINUITY ( VERTICAL

ASYMPTOTES AND HOLES FOR THE GRAPH OF EACH RATIONAL

FUNCTION).

)2)(1(

1

xx

xy

DESCRIBE THE ANY POINTS OF DISCONTINUITY ( VERTICAL

ASYMPTOTES AND HOLES FOR THE GRAPH OF EACH RATIONAL

FUNCTION).

)2(

)1)(2(

x

xxy

DESCRIBE THE ANY POINTS OF DISCONTINUITY ( VERTICAL

ASYMPTOTES AND HOLES FOR THE GRAPH OF EACH RATIONAL

FUNCTION).

)2)(1(

2

xx

xy