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This chapter delves into the transformations of sine and cosine functions, specifically focusing on how changes in coefficients affect amplitude, period, and phase shifts. When the amplitude (A) is modified, we see variations in the height of the graph. Negative coefficients cause reflections across the axes, while coefficients for the angle (B) manipulate the period of the functions. Furthermore, adding constants introduces vertical translations or displacements of the graphs. This exploration enhances comprehension of graph behavior in relation to function transformations.
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Chapter 4:Graphing & Inverse Functions 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) = sin x f (x) = ½sin x
f (x) = sin x f (x) = ¼sin x
f (x) = A sin x A indicates the amplitude A indicates the ??????????? the amplitude is A times larger than that of the basic sine curve (amp = 1)
f (x) = cosx f (x) = cos x
f (x) = cosx f (x) = 2cos x f (x) = 3cos x
f (x) = cosx f (x) = ½cos x f (x) = ¼cos x
Amplitude is the distance from the midline…so always positive. f (x) = A sin x or f (x) = A cosx A indicates the amplitude |A| indicates the amplitude the amplitude is A times larger than that of the basic sine or cosine curve the amplitude is |A| times larger than that of the basic sine or cosine curve the amplitude is A times larger than that of the basic sine 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 cosx If A is negative the graph is reflected across the x-axis.
f (x) = A sin x or f (x) = A cosx Domain: Range: Amplitude: Period:
f (x)=cosx f (x)=-3cosx
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
f (x) = sin Bx 2p ___ B indicates the period indicates the ????????? B indicates the ????????? the period of the function is the period of the basic curve divided by B (period = 2p)
f (x) = cosx f (x) = cos 2x f (x) = cos ½x
f (x) = sin Bx or f (x) = cos Bx f (x) = sin Bx Period for tangent and cotangent will be based on its period of π. 2p ___ B indicates the period the period of the function is the period of the basic curve divided by B (period = 2p)
f (x) = sin x SINE odd function f(-x) = - f(x) origin symmetry f (x) = sin -x Also graph of: f (x) = -sin x !
f (x) = cosx COSINE even function f(-x) = f(x) y-axis symmetry f (x) = cos -x Also graph of: f (x) = cosx !
f (x) = sin Bx or f (x) = cos Bx If B is negative the graph is reflected across the y-axis.
f(x)=sin Bxor f(x)=cosBx Domain: Range: Amplitude: Period: B < 0 means y-axis reflection
f(x)= sinx f(x)= sinx + 1
f(x)= sinx f(x)= sinx - 2
f(x)= cosx f(x)= cosx – 3
f (x) = sin x + D or f (x) = cosx + D D indicates displacement. The displacement is a vertical translation (shift) upward for D > 0 and downward for D < 0
f(x)=sinx f(x)=sin(x+ )
f(x)=cosx f(x)=cos(x- )
f(x)=cosx Phase shift is NOT p! Phase shift is NOT p! Coefficients affect the phase shift! = cos[2(x- )] f(x)=cos(2x- )
f(x)=sinx Alternate method: (negative means left) = sin[ (x+ )] f(x)=sin( x+ )
f(x)=sinx = sin[ (x+ )] f(x)=sin( x+ )
= sin(Bx+C) = cos(Bx+C) = sin[B(x+C)] = cos[B(x+C)] f (x) f (x) or -C ___ B indicates phase shift. -C The phase shift is a horizontal translation left for C > 0 and right for C < 0
Amplitude: A A < 0 reflect x Period: B < 0 reflect y 2p ___ B f (x)= A sin[B(x + C )] + D f (x)= Acos[B(x + C )] + D Phase shift: Displacement: D -C D > 0 up D < 0 down C > 0 left C < 0 right
f(x)=2sin(x-π)-2 Looks like sine reflected also…
