Consider the graph of y = \sin x.
Sketch the resulting graph when \\y = \sin x is translated vertically 4 units down.
State the equation of the translated function.
Consider the graph of y = \cos x.
Sketch the resulting graph when \\y = \cos x is translated vertically 2 units down.
State the equation of the translated function.
Complete the table of values for y = \sin x + 4:
| x | 0 | \dfrac{\pi}{2} | \pi | \dfrac{3\pi}{2} | 2\pi |
|---|---|---|---|---|---|
| y |
Sketch the graph of the function on the domain 0 \leq x \leq 2\pi.
Describe the transformation of the graph of the function y = \sin x that results in the graph of the function y = \sin x + 4.
State the maximum value of y = \sin x + 4.
State the minimum value of y = \sin x + 4.
For each of the following functions:
State the amplitude of the function.
Sketch the graph of the function.
y = 4 \cos x
y = 4 \sin x
y = - 5 \cos x
y = \dfrac{4}{3} \cos x
Consider the functions f \left( x \right) = \cos x and g \left( x \right) = \cos \left(\dfrac{x}{3}\right).
State the period of f \left( x \right) in radians.
Determine the period of g \left( x \right) in radians.
Complete the table of values for g \left( x \right):
| x | 0 | \pi | 2\pi | 3\pi | 4\pi | 5\pi | 6\pi |
|---|---|---|---|---|---|---|---|
| g\left(x\right) |
Describe the transformation of the graph of f \left( x \right) that results in the graph of g \left( x \right).
Sketch the graph of g \left( x \right) on the domain 0 \leq x \leq 6\pi .
Is the amplitude of g \left( x \right) different to the amplitude of f \left( x \right)?
Sketch the graphs of the following functions on the domain -2\pi \leq x \leq 2\pi:
y = 3 \cos x
y = - 4 \sin x
y = \sin 2 x
y = - \cos 3 x
y = \sin x + 4
y = \sin x - 2
y = \cos \left(x - \dfrac{\pi}{2}\right)
y = \sin \left(x + \dfrac{\pi}{2}\right)
Consider the functions f \left( x \right) = \cos x and g \left( x \right) = \cos 4 x.
State the period of f \left( x \right), in radians.
Complete the table of values for g \left( x \right):
| x | 0 | \dfrac{\pi}{8} | \dfrac{\pi}{4} | \dfrac{3\pi}{8} | \dfrac{\pi}{2} | \dfrac{5\pi}{8} | \dfrac{3\pi}{4} | \dfrac{7\pi}{8} | \pi |
|---|---|---|---|---|---|---|---|---|---|
| g(x) |
State the period of g \left( x \right), in radians.
Describe the transformation required to obtain the graph of g \left( x \right) from f \left( x \right).
Sketch the graph of g \left( x \right) for 0 \leq x \leq \pi.
Consider the table of values for the first complete cycle of the graph y=\sin x for x \geq 0.
| x | 0 | \dfrac{\pi}{2} | \pi | \dfrac{3\pi}{2} | 2\pi |
|---|---|---|---|---|---|
| \sin x | 0 | 1 | 0 | -1 | 0 |
Complete the table with values for the first complete cycle of the graph \\y = \sin \left(\dfrac{x}{4}\right) for x \geq 0.
| x | |||||
|---|---|---|---|---|---|
| \sin\left(\dfrac{x}{4}\right) | 0 | 1 | 0 | -1 | 0 |
Complete the table with values for the first complete cycle of the graph \\y = \sin 3x for x \geq 0.
| x | |||||
|---|---|---|---|---|---|
| \sin 3x | 0 | 1 | 0 | -1 | 0 |
Hence, state the period of y = \sin \left(\dfrac{x}{4}\right).
Hence, state the period of y = \sin 3x.
Consider the function y = 4 - 3 \sin x.
Find the maximum value of the function.
Find the minimum value of the function.
Consider the function f \left( x \right) = \cos 5 x.
Determine the period of the function, in radians.
Determine the exact number of cycles the curve completes in 15 radians.
Find the maximum value of the function.
Find the minimum value of the function.
Sketch the graph the function for 0 \leq x \leq \dfrac{4}{5} \pi.
For each of the following functions:
State the amplitude of the function.
Find the period of the function, in degrees.
Sketch the graph of the function on the domain -360 \degree \leq x \leq 360 \degree.
y = \sin \left( \dfrac{2}{3} x\right)
y = 3 \sin 2 x
For each of the following functions:
Determine the period of the function, in degrees.
State the amplitude of the function.
Find the maximum value of the function.
Find the minimum value of the function.
Sketch the graph of the function on the domain -360 \degree \leq x \leq 360 \degree.
y = 3 \sin x - 3
y = 3 \sin 2 x - 2
For each of the following functions:
State the amplitude of the function.
Determine the period of the function, in radians.
Sketch the graph of the function.
y = \cos \dfrac{1}{3} x
y = \cos \dfrac{4}{3} x
y = 5 \cos \dfrac{1}{2} x
y = 2 \sin 3 x
y = - 5 \sin 3 x
For each of the following functions:
Determine the period of the function, in radians.
State the amplitude of the function.
Determine the maximum value of the function.
Determine the minimum value of the function.
Sketch the graph of the function.
y = \sin \pi x
y = 3 \sin x + 2
y = \sin 2 x + 2
y = 2 \sin 3 x - 2
y = \cos 3 x + 2
y = 3 \sin \left(x - \dfrac{\pi}{3}\right) + 2
y = 2 \cos \left(x - \dfrac{\pi}{2}\right) + 3
For each of the following functions:
State the amplitude of the function.
Determine the phase shift of the function, in radians. Use a positive value to represent a shift to the right, and a negative value to represent a shift to the left.
Sketch the graph of the function.
y = 4 \sin \left(x - \pi\right)
y = \sin \left(x - \dfrac{\pi}{2}\right)
y = 5 \cos \left(x - \dfrac{\pi}{3}\right)
y = - 4 \cos \left(x + \dfrac{\pi}{2}\right)
For each of the following functions:
Determine the period of the function, in radians.
Determine the phase shift of the function, in radians.
Sketch the graph of the function for -\pi \leq x \leq \pi.
y = \cos \left( 3 \left( x + \dfrac{\pi}{3} \right)\right)
y = \sin \left( 2 x - \dfrac{2 \pi}{3}\right)
For each of the following functions:
State the amplitude of the function.
Determine the period of the function, in radians.
Determine the phase shift of the function, in radians.
State the range of the function.
Sketch the graph of the function for -\pi \leq x \leq \pi.
y = - 3 \sin \left( 4 \left(x - \dfrac{\pi}{4} \right) \right)
y = - 2 \sin \left( 4 x - \pi\right)
Consider the functions y = 2 \cos \left(x - \dfrac{\pi}{3}\right) and y = \sin \left(\dfrac{x}{4}\right).
Sketch the graphs of the two functions on the same set of axes on the domain \\0 \leq x \leq 4 \pi.
Hence determine the number of solutions on the domain 0 \leq x \leq 4 \pi, for the equation: \\2 \cos \left(x - \dfrac{\pi}{3}\right) - \sin \left(\dfrac{x}{4}\right) = 0