7.12 Period and phase shifts of trigonometric graphs
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 function y = 2 \cos 3 x.
State the amplitude of the function.
Find the period of the function in radians.
Sketch a graph of the function for -\pi \leq x \leq \pi.
For each of the following functions:
State the amplitude.
Find the period in radians.
Sketch a graph of the function for 0 \leq x \leq 2\pi.
Consider the function y = \sin \left( \dfrac{2x}{3} \right).
State the amplitude of the function.
Find the period of the function in degrees.
Sketch a graph of the function for 0\degree \leq x \leq 720 \degree.
A table of values for the the first period of the graph y=\sin x for x \geq 0 is given in the first table on the right:
Complete the second table given with equivalent values for x in the the first period of the graph y = \sin \left(\dfrac{x}{4}\right) for \\x \geq 0.
Hence, state the period of y = \sin \left(\dfrac{x}{4}\right).
| x | 0 | \dfrac{\pi}{2} | \pi | \dfrac{3\pi}{2} | 2\pi |
|---|---|---|---|---|---|
| \sin x | 0 | 1 | 0 | -1 | 0 |
| x | |||||
|---|---|---|---|---|---|
| \sin\left(\dfrac{x}{4}\right) | 0 | 1 | 0 | -1 | 0 |
Consider functions of the form y=\tan bx.
Complete the given table identifying the periods of the functions.
State the period of y = \tan b x.
As the value of b in \tan b x increases, does the period become shorter or longer?
| Function | Period |
|---|---|
| \tan x | \pi |
| \tan 2x | |
| \tan 3x | |
| \tan 4x |
State the period of the following functions:
The functions f \left( x \right) and g \left( x \right) = f \left( kx \right) have been graphed on the same set of axes below.
Describe the transformation required to obtain the graph of g\left(x\right) from the graph of f \left( x \right).
Find the value of k.
Consider the function f\left(x\right)=\tan x and functions of the form g\left(x\right)=\tan bx.
Hence, sketch each of the following funtions for 0 \leq x \leq 2\pi:
For each of the following functions:
Find the y-intercept.
Find the period of the function in radians.
Find the distance between the asymptotes of the function.
State the first asymptote of the function for x \geq 0
State the first asymptote of the function for x \leq 0
Sketch a graph of the function for -\pi \leq x \leq \pi.
Consider the equation y = \tan 9 x.
State the period of the equation in radians.
Sketch the graph of the equation y = \tan 9 x for 0 \leq x \leq \pi
Consider the function y = \tan 7 x.
Complete the given table of values for the function.
Graph the function for -\dfrac{5\pi}{28}\leq x \leq \dfrac{5\pi}{28}.
| x | -\dfrac{\pi}{28} | 0 | \dfrac{\pi}{28} | \dfrac{3\pi}{28} | \dfrac{\pi}{7} | \dfrac{5\pi}{28} |
|---|---|---|---|---|---|---|
| y |
A function in the form f \left( x \right) = \tan b x has adjacent x-intercepts at x = \dfrac{13 \pi}{6} and x = \dfrac{9 \pi}{4}.
State the equation of the asymptote lying between the two x-intercepts.
Find the period of the function.
State the equation of the function.
Consider the graph of f \left( x \right) = \tan \left( \alpha x\right), and the graph of g \left( x \right) = \tan \left( \beta x\right) below:
Which is greater: \alpha or \beta? Explain your answer.
A table of values for the the first period of the graph y=\tan x for x \geq 0 is given in the first table on the right:
Complete the second table given with equivalent values for x in the the first period of the graph \\y = \tan \left(\dfrac{x}{6}\right) for x \geq 0.
Hence, state the period of \\y = \tan \left(\dfrac{x}{6}\right).
| x | 0 | \dfrac{\pi}{4} | \dfrac{\pi}{2} | \dfrac{3\pi}{4} | \pi |
|---|---|---|---|---|---|
| \tan x | 0 | 1 | \text{undefined} | -1 | 0 |
| x | |||||
|---|---|---|---|---|---|
| \tan\left(\dfrac{x}{6}\right) | 0 | 1 | \text{undefined} | -1 | 0 |
State the first three asymptotes of the function for x \geq 0.
State the first asymptote of the function for x \leq 0.
For each of the following functions of the form y=A \tan bx:
Find the value of the function at x=\dfrac{\pi}{4b}.
Find the period of the function in radians.
State the first asymptote of the function for x \geq 0
State the first asymptote of the function for x \leq 0
Sketch a graph of the function for -2\pi \leq x \leq 2\pi.
Consider the function y = \cos 3 x + 2.
Find the period of the function, giving your answer in radians.
State the amplitude of the function.
Find the maximum value of the function.
Find the minimum value of the function.
Sketch a graph of the function for 0 \leq x \leq 2\pi.
For each of the following functions:
State the domain of the function.
State the range of the function.
Sketch a graph of the function for -\pi \leq x \leq \pi.
y = \sin 2 x - 2
y = - 5 \sin 2 x
y = \sin \left(\dfrac{x}{3}\right) + 5
y = \cos \left(\dfrac{x}{2}\right) - 3
For each of the following graphs, find the equation of the function given that it is of the form y = a \sin b x or y = a \cos b x, where b is positive:
State whether the following functions represent a change in the period from the function y = \sin x:
y = \sin \left( 5 x\right)
y = 5 \sin x
y = \sin \left( \dfrac{x}{5} \right)
y = \sin x + 5
Consinder the function f \left( x \right) = 3 \tan \left( 3 x\right) + 2.
State the domain of f \left( x \right).
State the range of f \left( x \right).
The graph of f \left( x \right) has its domain restricted to \left( - \dfrac{\pi}{6} , 0\right], state the range of the restricted graph.