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Analyzing Trigonometric Graphs

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Axis of Symmetry in Sine Graphs

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Sine graphs are symmetric about the y-axis when they are not phase shifted. This means f(x)=f(x)f(-x) = -f(x) for all xx in the graph of y=sin(x)y = \sin(x).

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Frequency of a Trigonometric Graph

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Frequency is the number of cycles the graph completes in an interval of 2π2\pi. For y=Asin(Bx)y = A\sin(Bx) or y=Acos(Bx)y = A\cos(Bx), the frequency is B/(2π)|B|/(2\pi).

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Phase Shift of a Sine or Cosine Graph

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Phase shift denotes the horizontal displacement of the graph. For y=Asin(B(xC))y = A\sin(B(x - C)) or y=Acos(B(xC))y = A\cos(B(x - C)), the phase shift is CC units to the right if C>0C > 0 and to the left if C<0C < 0.

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Period of a Sine or Cosine Graph

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The period is the distance along the x-axis to complete one cycle of the graph. For y=Asin(Bx)y = A\sin(Bx) or y=Acos(Bx)y = A\cos(Bx), the period is 2πB\frac{2\pi}{|B|}.

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Graphing y=tan(x)y = \tan(x)

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The tangent graph has undefined values at x=(2n+1)π2x = (2n + 1)\frac{\pi}{2}, where nn is an integer, resulting in vertical asymptotes. The period of the tangent function is π\pi.

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Graphing y=cot(x)y = \cot(x)

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The cotangent graph has undefined values at x=nπx = n\pi, where nn is an integer, leading to vertical asymptotes. The period of the cotangent function is also π\pi.

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Amplitude of a Sine or Cosine Graph

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Amplitude corresponds to the maximum deviation from the graph's central axis. For the functions y=Asin(Bx)y = A\sin(Bx) or y=Acos(Bx)y = A\cos(Bx), the amplitude is A|A|.

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Reflection Over the x-axis

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A reflection over the x-axis changes the sign of the amplitude. For y=Asin(Bx)y = -A\sin(Bx) or y=Acos(Bx)y = -A\cos(Bx), the graph is reflected over the x-axis.

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Vertical Shift of a Sine or Cosine Graph

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Vertical shift refers to the upward or downward translation of the graph. For y=Asin(Bx)+Dy = A\sin(Bx) + D or y=Acos(Bx)+Dy = A\cos(Bx) + D, the graph is shifted DD units up for D>0D > 0 or down for D<0D < 0.

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Axis of Symmetry in Cosine Graphs

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Cosine graphs are symmetric about the y-axis, which implies f(x)=f(x)f(-x) = f(x) for all xx in the graph of y=cos(x)y = \cos(x), making it an even function.

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Determining the Equation from a Trigonometric Graph

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To find the equation from a graph, identify the amplitude (A), period (2π/B2\pi/B), phase shift (C), and vertical shift (D), then use the sine or cosine model y=Asin(B(xC))+Dy = A\sin(B(x-C)) + D or y=Acos(B(xC))+Dy = A\cos(B(x-C)) + D.

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Transformations involving Multiple Components

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Complex transformations like y=Asin(B(xC))+Dy = A\sin(B(x - C)) + D involve stretching/compressing vertically by AA, horizontally by BB, shifting horizontally by CC, and vertically by DD. Analyze these separately for clarity.

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