Chapter 4: Graphing & Inverse Functions Sections 4.2, 4.3, & 4.5 Transformations Sections 4.2, 4.3, & 4.5 Transformations

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

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 amplitude the amplitude is | A | times larger than that of the basic sine or cosine curve

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 negative the graph is reflected across the x -axis.

f ( x ) = A sin x or f ( x ) = A cos x Domain: Range: Amplitude: Period:

f ( x ) = cos x f ( x ) = -3cos x

f ( x ) = sin x f ( x ) = sin 2 x

f ( x ) = sin 2 x f ( x ) = sin x

f ( x ) = sin 4 x

f ( x ) = sin x f ( x ) = sin ½ x

f ( x ) = sin x f ( x ) = sin ¼ x

indicates the ????????? indicates the period B indicates the ????????? f ( x ) = sin Bx the period of the function is the period of the basic curve divided by B (period = 2  ) 22 B ___

f ( x ) = cos x f ( x ) = cos ½ x f ( x ) = cos 2 x

indicates the period f ( x ) = sin Bx or f ( x ) = cos Bx the period of the function is the period of the basic curve divided by B (period = 2  ) 22 B ___ Period for tangent and cotangent will be based on its period of π. f ( x ) = sin Bx

f ( x ) = sin x f ( x ) = sin - x SINE odd function f (- 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 negative the graph is reflected across the y -axis.

f ( x )=sin Bx or f ( x )=cos Bx Domain: Range: Amplitude: Period: B < 0 means y -axis reflection

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 + D or f ( x ) = cos x + D D indicates displacement. The displacement is a vertical translation (shift) upward for D > 0 and downward for D < 0

f ( x )=sin x f ( x )=sin( x+ )

f ( x )=cos x f ( x )=cos( x- )

f ( x )=cos x f ( x )=cos(2 x- ) = cos[2( x- )] Phase shift is NOT  ! Coefficients affect the phase shift!

f ( x )=sin x f ( x )=sin( x+ ) = sin[ ( x+ )] Alternate method: (negative means left)

f ( x )=sin x f ( x )=sin( x+ ) = sin[ ( x+ )]

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-C or f ( x ) -C-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-C Amplitude: A A < 0 reflect x ___ 22 B Displacement: D D > 0 up D < 0 down

f ( x )= 3sin(2 x )+1

f ( x )= 2sin( x - π ) -2 Looks like sine reflected also…

=-1cos2( x - π/2 ) +3 f ( x )=-cos(2 x - π ) +3

f ( x )= 2 sin [ ½(x + 0)] - 2 f ( x )=2sin(½ x)-2 What is the equation? f ( x )= A sin [ B(x + C )] + D f ( x )= A sin [ B(x + C )] - 2 f ( x )= 2 sin [ B(x + C )] - 2 f ( x )= 2 sin [ ½(x + C )  - 2 Could also be written with shifted cosine:

MODE On a TI-84 calculator: Function (Func Par Pol Seq) Radian (Radian Degree) y =y = y = 2 sin ( (1/2) x ) - 2 WINDOW X min = -4 π X max = 4 π X scl = π/2 Y min = -4 Y max = 4 Y scl = 1 SCALE determines the location of the tic marks.

On a TI-Nspire calculator: 5 Settings 2: Document Settings Angle: Radian y = 2 sin ( (1/2) x ) - 2 XMin: -4 π XMax: 4 π XScale: π/2 YMin: -4 YMax: 4 YScale: 1 B Graph 4: Window/Zoom 1: Window Settings 3: Graph Entry 1: Function

The End