Step By Step Calculus » 6.3 - Sketching Trigonometric Functions

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Synopsis
Graphs of Sine and Cosine: To sketch the graph of y=a\sin (bx+c)+dy=a\sin (bx+c)+d for real constants a,b,c,da,b,c,d, we use the following procedure (we call it FART) and the graph of \sin x\sin x as the base graph for Sine.
Step 1 ( Form) Convert to standard form y=A\sin B(x+C)+Dy=A\sin B(x+C)+D with B>0B>0. If b>0b>0, then A=a, B=b, C=\frac{c}{b}, D=dA=a, B=b, C=\frac{c}{b}, D=d. If b<0b<0, then A=-a, B=-b, C=\frac{c}{b}, D=dA=-a, B=-b, C=\frac{c}{b}, D=d by the oddness of Sine.
Step 2( Assign) Note amplitude |A||A| (vertical stretch/compress), period \frac{2\pi}{B}\frac{2\pi}{B} (horizontal stretch/compress), phase shift CC (horizontal shift to left/right) and vertical shift DD (up/down). Draw the graph of Sine with amplitude |A||A| and period \frac{2\pi}{B}\frac{2\pi}{B} from the base graph for Sine that has amplitude 11 and period 2\pi2\pi.
Step 3( Reflect) If A<0A<0, reflect the resulting graph in the xx-axis.
Step 4( Translate) If C>0C>0, shift the graph horizontally left CC units. If C<0C<0, shift the graph horizontally right |C||C| units. If C=0C=0, no horizontal shifting is needed.
If D>0D>0, shift the graph vertically up DD units. If D<0D<0, shift the graph vertically down |D||D| units. If D=0D=0, no vertical shifting is needed.
It will be similar for y=a\cos (bx+c)+dy=a\cos (bx+c)+d except in the Step 1 where, because of the evenness of cosine, we will have A=a, B=b, C=\frac{c}{b}, D=dA=a, B=b, C=\frac{c}{b}, D=d if bb is positive and A=a, B=-b, C=\frac{c}{b}, D=dA=a, B=-b, C=\frac{c}{b}, D=d if bb is negative. We will use the graph of \cos x \cos x as the base graph for Cosine in Step 2.
Graphs of Secant and Cosecant: \sec x\sec x has vertical asymptotes x=\frac{\pi}{2}\pm n\pix=\frac{\pi}{2}\pm n\pi for n\in\mathbb{N}_0n\in\mathbb{N}_0 and \csc x\csc x has vertical asymptotes x=\pm n\pix=\pm n\pi for n\in\mathbb{N}_0n\in\mathbb{N}_0.
To draw the graph of a function of the form y=a\sec (bx+c)+dy=a\sec (bx+c)+d or y=a\csc (bx+c)+dy=a\csc (bx+c)+d with a,b,c,d\in\mathbb{R}a,b,c,d\in\mathbb{R}, we use the FART procedure described earlier with the following things in mind. In Step 1, when converting to standard form, we have to use the evenness of \sec \sec and oddness of \csc\csc if b<0b<0. In Step 2, we use the value of AA to refer to vertical stretch instead of amplitude since these graphs increase and decrease forever. In Step 4, we have to remember that the asymptotes are part of the graph and must be shifted.
Graphs of Tangent and Cotangent: \tan x\tan x has vertical asymptotes x=\frac{\pi}{2}\pm n\pix=\frac{\pi}{2}\pm n\pi for n\in\mathbb{N}_0n\in\mathbb{N}_0 and \cot x\cot x has vertical asymptotes x=\pm n\pix=\pm n\pi for n\in\mathbb{N}_0n\in\mathbb{N}_0.
To draw the graph of a function of the form y=a\tan (bx+c)+dy=a\tan (bx+c)+d or y=a\cot (bx+c)+dy=a\cot (bx+c)+d with a,b,c,d\in\mathbb{R}a,b,c,d\in\mathbb{R}, we use the FART procedure described earlier with the following things in mind. In Step 1, when converting to standard form, we have to use the oddness of \tan\tan and \cot\cot if b<0b<0. In Step 2, we use the value of AA to refer to vertical stretch instead of amplitude since these graphs increase and decrease forever. The period of \tan\tan and \cot\cot is \dfrac{\pi}{B}\dfrac{\pi}{B} not \dfrac{2\pi}{B}\dfrac{2\pi}{B}. In Step 4, we have to remember that the asymptotes need to be shifted when performing shift operation.