The Reciprocal Function Family and rational functions and their...
Transcript of The Reciprocal Function Family and rational functions and their...
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