Test Objectives
  • Demonstrate the ability to sketch the graph of sine
  • Demonstrate the ability to sketch the graph of cosine
  • Demonstrate the ability to find the period of the sine or cosine function
  • Demonstrate the ability to find the amplitude of the sine or cosine function
  • Demonstrate the ability to find the phase shift of the sine or cosine function
  • Demonstrate the ability to find the vertical shift of the sine or cosine function
Graphing Sine & Cosine Practice Test:

#1:

Instructions: Find the amplitude and period in radians, sketch the graph.

$$a)\hspace{.1em}f(x)=\text{sin}\hspace{.1em}4x$$

$$b)\hspace{.1em}f(x)=3 \hspace{.1em}\text{sin}\frac{x}{4}\hspace{.1em}$$


#2:

Instructions: Find the amplitude, period, and phase shift in radians, sketch the graph.

$$a)\hspace{.1em}f(x)=2 \hspace{.1em}\text{cos}\left(3x - \frac{π}{2}\right)$$

$$b)\hspace{.1em}f(x)=\text{cos}\left(3x - \frac{5π}{6}\right)$$


#3:

Instructions: Find the amplitude, period, and phase shift in radians, sketch the graph.

$$a)\hspace{.1em}f(x)=3 \hspace{.1em}\text{sin}\left(4x + \frac{π}{6}\right)$$

$$b)\hspace{.1em}f(x)=2 \hspace{.1em}\text{sin}\left(2x - \frac{3π}{4}\right)$$


#4:

Instructions: Find the amplitude, period, and phase shift in radians. Find the vertical shift, sketch the graph.

$$a)\hspace{.1em}f(x)=3 \hspace{.1em}\text{sin}\left(\frac{x}{2}- \frac{π}{3}\right) + 2$$

$$b)\hspace{.1em}f(x)=3 \hspace{.1em}\text{cos}\left(4x - \frac{2π}{3}\right) - 1$$


#5:

Instructions: Find the amplitude, period, and phase shift in radians. Find the vertical shift, sketch the graph.

$$a)\hspace{.1em}f(x)=4 \hspace{.1em}\text{cos}\left(4x + \frac{3π}{2}\right) + 2$$

$$b)\hspace{.1em}f(x)=2 \hspace{.1em}\text{cos}\left(x + \frac{3π}{4}\right) - 2$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}\text{Amplitude:}\hspace{.1em}1, \text{Period:}\hspace{.1em}\frac{π}{2}$$ Graphing f(x)=sin 4x

$$b)\hspace{.1em}\text{Amplitude:}\hspace{.1em}3, \text{Period:}\hspace{.1em}8 π$$ Graphing f(x)=3 sin x/4


#2:

Solutions:

$$a)\hspace{.1em}\text{Amplitude:}\hspace{.1em}2, \text{Period:}\hspace{.1em}\frac{2π}{3}$$ $$\text{Phase Shift:}\hspace{.1em}\text{Right}\hspace{.1em}\frac{π}{6}$$ Graphing f(x)=2 cos (3x - π/2)

$$b)\hspace{.1em}\text{Amplitude:}\hspace{.1em}1, \text{Period:}\hspace{.1em}\frac{2π}{3}$$ $$\text{Phase Shift:}\hspace{.1em}\text{Right}\hspace{.1em}\frac{5π}{18}$$ Graphing f(x)=cos (3x - 5π/6)


#3:

Solutions:

$$a)\hspace{.1em}\text{Amplitude:}\hspace{.1em}3, \text{Period:}\hspace{.1em}\frac{π}{2}$$ $$\text{Phase Shift:}\hspace{.1em}\text{Left}\hspace{.1em}\frac{π}{24}$$ Graphing f(x)=3 sin (4x + π/6)

$$b)\hspace{.1em}\text{Amplitude:}\hspace{.1em}2, \text{Period:}\hspace{.1em}π$$ $$\text{Phase Shift:}\hspace{.1em}\text{Right}\hspace{.1em}\frac{3π}{8}$$ Graphing f(x)=2 sin (2x - 3π/4)


#4:

Solutions:

$$a)\hspace{.1em}\text{Amplitude:}\hspace{.1em}3, \text{Period:}\hspace{.1em}4π$$ $$\text{Phase Shift:}\hspace{.1em}\text{Right}\hspace{.1em}\frac{2π}{3}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Up}\hspace{.1em}2 \hspace{.1em}\text{units}$$ Graphing f(x)=3 sin (x/2 - π/3) + 2

$$b)\hspace{.1em}\text{Amplitude:}\hspace{.1em}3, \text{Period:}\hspace{.1em}\frac{π}{2}$$ $$\text{Phase Shift:}\hspace{.1em}\text{Right}\hspace{.1em}\frac{π}{6}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Down}\hspace{.1em}1 \hspace{.1em}\text{unit}$$ Graphing f(x)=3 cos (4x - 2π/3) - 1


#5:

Solutions:

$$a)\hspace{.1em}\text{Amplitude:}\hspace{.1em}4, \text{Period:}\hspace{.1em}\frac{π}{2}$$ $$\text{Phase Shift:}\hspace{.1em}\text{Left}\hspace{.1em}\frac{3π}{8}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Up}\hspace{.1em}2 \hspace{.1em}\text{units}$$ Graphing f(x)=4 cos (4x + 3π/2) + 2

$$b)\hspace{.1em}\text{Amplitude:}\hspace{.1em}2, \text{Period:}\hspace{.1em}2π$$ $$\text{Phase Shift:}\hspace{.1em}\text{Left}\hspace{.1em}\frac{3π}{4}$$ $$\text{Vertical Shift:}\hspace{.1em}\text{Down}\hspace{.1em}2 \hspace{.1em}\text{units}$$ Graphing f(x)=2 cos (x + 3π/4) - 2