Question 1: Sketch the graphs of the following functions:

\displaystyle \text{i) } f(x) = 2 \sin x , 0 \leq x \leq \pi \hspace{1.0cm} \text{ii) } f(x) = 3 \sin \Big( x -  \frac{\pi}{4}  \Big), 0 \leq x \leq  \frac{5\pi}{4}

\displaystyle \text{iii) } f(x) = 2 \sin 3x, 0 \leq x \leq  \frac{2\pi}{3} \hspace{1.0cm}  \text{iv) } f (x) = 2 \sin \Big( 2x -  \frac{\pi}{3}  \Big), 0 \leq x \leq  \frac{7\pi}{5}

\displaystyle \text{v) } f(x) = 4 \sin 3 \Big( x -  \frac{\pi}{4}  \Big), 0 \leq x \leq 2\pi \hspace{0.5cm}  \text{vi) } f(x) = \sin \Big(  \frac{x}{2}  -  \frac{\pi}{4}  \Big), 0 \leq x \leq 4\pi

\displaystyle \text{vii) } f(x) = \sin^2 x , 0 \leq x \leq 2\pi ,

\displaystyle \text{viii) } g(x) = |\sin x |, 0 \leq x \leq 2\pi

\displaystyle \text{ix) } f(x) = 2 \sin \pi x, 0 \leq x \leq 2

Answer:

i) \displaystyle f(x) = 2 \sin x , 0 \leq x \leq \pi

611

ii) \displaystyle f(x) = 3 \sin \Big( x -  \frac{\pi}{4}  \Big), 0 \leq x \leq  \frac{5\pi}{4}

612

iii) \displaystyle f(x) = 2 \sin 3x, 0 \leq x \leq  \frac{2\pi}{3}

613

iv) \displaystyle f (x) = 2 \sin \Big( 2x -  \frac{\pi}{3}  \Big), 0 \leq x \leq  \frac{7\pi}{5}

614

v) \displaystyle f(x) = 4 \sin 3 \Big( x -  \frac{\pi}{4}  \Big), 0 \leq x \leq 2\pi  

615  

vi) \displaystyle f(x) = \sin \Big(  \frac{x}{2}  -  \frac{\pi}{4}  \Big), 0 \leq x \leq 4\pi

616

vii) f(x) = \sin^2 x , 0 \leq x \leq 2\pi ,

617

viii) g(x) = |\sin x |, 0 \leq x \leq 2\pi

618

ix) f(x) = 2 \sin \pi x, 0 \leq x \leq 2

619

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Question 2:

\displaystyle \text{i) } f (x) = \sin x , \ \ g(x) = \sin \Big( x +  \frac{\pi}{4}  \Big) \hspace{1.0cm} \text{ii) } f (x) = \sin x , \ \ g(x) = \sin 2x

\displaystyle \text{iii) } f (x) = \sin 2x , \ \ g(x) = 2 \sin x \hspace{1.0cm} \text{iv) } f (x) = \sin  \frac{x}{2}  , \ \ g(x) = \sin x

Answer:

\displaystyle \text{i) } f (x) = \sin x \text{ (Red Line) }  g(x) = \sin \Big( x +  \frac{\pi}{4}  \Big) \text{ (Blue Line) }

6121

\displaystyle \text{ii) } f (x) = \sin x \text{ (Red Line) }  g(x) = \sin 2x \text{ (Blue Line) }

6122

\displaystyle \text{iii) } f (x) = \sin 2x \text{ (Red Line) }  g(x) = 2 \sin x \text{ (Blue Line) }    

6123

\displaystyle \text{iv) } f (x) = \sin  \frac{x}{2} \text{ (Red Line) }  g(x) = \sin x \text{ (Blue Line) }

6124