Chapter 4: Graphing & Inverse Functions Sections 4.2, 4.3, & 4.5 Transformations Sections 4.2, 4.3,...
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Transcript of Chapter 4: Graphing & Inverse Functions Sections 4.2, 4.3, & 4.5 Transformations Sections 4.2, 4.3,...
Chapter 4:Graphing &
Inverse Functions
Chapter 4:Graphing &
Inverse Functions
Sections 4.2, 4.3, & 4.5
Transformations
Sections 4.2, 4.3, & 4.5
Transformations
What happens when the
function has a coefficient?
f (x) = sin x
f (x) = sin x
f (x) = 2sin x
f (x) = sin x
f (x) = 3sin x
f (x) = sin x
f (x) = 4sin x
f (x) = ½sin x
f (x) = sin x
f (x) = ¼sin x
f (x) = sin x
A indicates the ???????????the amplitude is A times larger than that of the basic sine curve (amp = 1)
A indicates the amplitude
f (x) = A sin x
What about
cosin
e?
f (x) = cos x
f (x) = cos x f (x) = cos x
f (x) = cos x
f (x) = 3cos x
f (x) = 2cos x
f (x) = cos x
f (x) = ¼cos x
f (x) = ½cos x
the amplitude is A times larger than that of the basic sine curve
the amplitude is A times larger than that of the basic sine or cosine curve
f (x) = A sin x or f (x) = A cos x A indicates the amplitude
Amplitude is the distance
from the midline…so
always positive.
|A| indicates the amplitudethe amplitude is |A| times larger than that of the basic sine or cosine curve
What about
negative
coeffi
cients?
f (x) = sin x
f (x) = -sin x
f (x) = cos x
f (x) = -cos x
f (x) = A sin x or
f (x) = A cos x
If A is negativethe graph is reflected
across the x-axis.
f (x) = A sin x or
f (x) = A cos x
Domain: Range:Amplitude:Period:
all A y A
A2
f (x) = cos x
f (x) = -3cos x
What happens when the ANGLE has a coefficient?
f (x) = sin x
f (x) = sin 2x
f (x) = sin 2x f (x) = sin x
f (x) = sin x
f (x) = sin 4x
f (x) = sin x
f (x) = sin ½x
f (x) = sin x
f (x) = sin ¼x
indicates the ?????????indicates the periodB indicates
the ?????????
f (x) = sin Bx
the period of the function is the period of the basic curve divided by B (period = 2p)
2pB
___
f (x) = cos x
f (x) = cos ½x
f (x) = cos 2x
indicates the period
f (x) = sin Bx orf (x) = cos Bx
the period of the function is the period of the basic curve divided by B (period = 2p)
2pB
___
Period for tangent and
cotangent will be based on
its period of π.
f (x) = sin Bx
What about
negative
coeffi
cients
of t
he
angle?
f (x) = sin x
f (x) = sin -x
SINE odd functionf(-x) = - f(x)
origin symmetry
Also graph of: f (x) = -sin x !
f (x) = cos x
f (x) = cos -x
COSINE even function
f(-x) = f(x)y-axis
symmetry
Also graph of: f (x) = cos x !
f (x) = sin Bx or
f (x) = cos Bx
If B is negativethe graph is reflected
across the y-axis.
f(x)=sin Bx or f(x)=cos Bx
Domain: Range:Amplitude:
Period:
all 1 1y
12
B
B < 0 means y-axis reflection
What happens when a constant is
added to the function?
f(x)= sin x
f(x)= sin x + 1
f(x)= sin x
f(x)= sin x - 2
f(x)= cos x
f(x)= cos x – 3
f (x) = sin x + Dor
f (x) = cos x + D
D indicates displacement.The displacement is a vertical translation (shift) upward for D > 0 and downward for D < 0
What happens when a constant is
added to the angle?
f(x)=sin x
f(x)=sin(x+ ) 4
f(x)=cos x
f(x)=cos(x- ) 3
2
f(x)=cos x
f(x)=cos(2x- )2
= cos[2(x- )]
Phase shift is NOT p!
Phase shift is NOT p!
Coefficients affect the phase shift!
f(x)=sin x
f(x)=sin( x+ )6
3
= sin[ (x+ )]
1
2
1
2
Alternate method:
10
2 6x
1
2 6x
12 2
2 6x
(negative means left)
3x
f(x)=sin x
f(x)=sin( x+ )6
3
= sin[ (x+ )]
1
2
1
2
indicates phase shift.
= sin(Bx+C)
= cos(Bx+C)The phase shift is a horizontal translation left for C > 0 and right for C < 0
= sin[B(x+C)]
= cos[B(x+C)]B
___-C
or f (x)
f (x)
-C
Period:
B < 0 reflect y
f (x)= A sin [B(x + C )] + D
f (x)= A cos [B(x + C )] + D
Phase shift:C > 0 left C < 0 right
-C
Amplitude: AA < 0 reflect x
___2pB
Displacement: DD > 0 up D < 0 down
f (x)= 3sin(2x)+1
f (x)= 2sin(x - π) -2Looks like sine reflected also…
2sin 2f x x
=-1cos2(x - π/2)
+3
f (x)=-cos(2x - π) +3
f (x)= 2 sin [½(x + 0)] - 2f(x)=2sin(½ x)-2
What is the equation?
f (x)= A sin [B(x + C )] + Df (x)= A sin [B(x + C )] - 2f (x)= 2 sin [B(x + C )] - 2f (x)= 2 sin [½(x + C )] - 2
Could also be written with shifted
cosine: 1
2cos 22
f x x
MODE
On a TI-84 calculator:
Function (Func Par Pol Seq)Radian (Radian Degree)
y = y = 2 sin ( (1/2) x ) - 2
WINDOW
Xmin = -4πXmax = 4πXscl = π/2
Ymin = -4Ymax = 4Yscl = 1
SCALE determines the location of the
tic marks.
On a TI-Nspire calculator:5 Settings2: Document SettingsAngle: Radian
y = 2 sin ( (1/2) x ) - 2
XMin: -4πXMax: 4πXScale: π/2
YMin: -4YMax: 4YScale: 1
B Graph
4: Window/Zoom1: Window Settings
3: Graph Entry1: Function
The End