6.02 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.
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.
Consider the function y = \sin \left(x + \dfrac{\pi}{2}\right).
Find the maximum value of the function.
Find the minimum value of the function.
Consider the function f \left( x \right) = \cos x and g \left( x \right) = \cos \left(x - \dfrac{\pi}{2}\right).
Complete the table of values for both functions:
Describe the transformation required to obtain the graph of g \left( x \right) from f \left( x \right).
Sketch a graph of g \left( x \right) for 0 \leq x \leq 2\pi.
| x | 0 | \dfrac{\pi}{2} | \pi | \dfrac{3\pi}{2} | 2\pi |
|---|---|---|---|---|---|
| f(x) | |||||
| g(x) |
For each of following functions:
State the amplitude of the function.
Describe the transformation required to obtain the graph of y =f\left(x\right) from the graph of y = \sin x.
Sketch a graph of the function for -\pi \leq x \leq \pi.
For each of following functions:
State the amplitude of the function.
Describe the transformation required to obtain the graph of y =f\left(x\right) from the graph of y = \cos x.
Sketch a graph of the function for -\pi \leq x \leq \pi.
Find the values of c in the region - 2 \pi \leq c \leq 2 \pi that make the graph of y = \sin \left(x - c\right) the same as the graph of y = \cos x.
Sketch the graph of the following for -2\pi \leq x \leq 2\pi.
True of false: The graph of f \left( x \right) = \tan \left(x - \dfrac{\pi}{6}\right) is a phase shift of y=\tan x by \dfrac{\pi}{6} units to the right.
For each of the functions below, state the horizontal translation required to obtain the graph of g \left( x \right) from the graph of f \left( x \right) = \tan x:
The function f \left( x \right) = \tan \left(x - \dfrac{\pi}{6}\right) is to be graphed on the interval \left[\dfrac{2 \pi}{3}, \dfrac{8 \pi}{3}\right].
Find the asymptotes of the function that occur on this interval.
Find the x-intercepts of the function that occur on this interval.
Sketch a graph of the function f \left( x \right) = \tan \left(x - \dfrac{\pi}{6}\right) over the interval \left[\dfrac{2 \pi}{3}, \dfrac{8 \pi}{3}\right].
For each of the following functions:
Find the y-intercept.
Find the period of the function.
Find the distance between the asymptotes of the function.
State the first asymptote of the function for x > 0.
State the first asymptote of the function for x \leq 0.
Sketch a graph the function for -\pi \leq x \leq \pi.
Consider the function f \left( x \right) = \tan \left(x - \dfrac{\pi}{7}\right).
Compared to the equation y = \tan x, state the phase shift of f \left( x \right).
State the equations of the first four asymptotes of f \left( x \right) to the right of the origin.
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
Sketch a graph of each of the following the functions for 0 \leq x \leq 2\pi:
Consider the graphs of y = \cos x and y = 3 \cos \left(x - \dfrac{\pi}{4}\right).
Describe the series of transformations required to obtain the graph of y = 3 \cos \left(x - \frac{\pi}{4} \right) from the graph of y=\cos x.
Consider the function f\left(x\right) = - 5 \sin \left(x + \dfrac{\pi}{3}\right).
Describe the series of transformations required to obtain the graph of f\left(x\right) from the graph of y=\sin x.
Sketch a graph of y = f \left(x\right) for -2\pi \leq x \leq 2\pi.
The graph of y = \cos x undergoes the series of transformations below:
The graph is reflected about the x-axis.
The graph is horizontally translated to the left by \dfrac{\pi}{6} radians.
The graph is vertically translated upwards by 5 units.
Find the equation of the transformed graph in the form y = a \cos \left(x + c\right) + d, where c is the lowest positive value in radians.
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:
The functions f \left( x \right) and g \left( x \right) = f \left( x - k \right) - j have been graphed on the same set of axes below:
Find the value of j.
Find the smallest positive value of k.
Describe the series of transformations required to obtain the graph of g\left(x\right) from the graph of f\left(x\right).
Describe the series of transformations required to obtain the graph of y = \sin \left(x - \dfrac{\pi}{4}\right) + 2 from the graph of y = \sin \left(x\right).
State whether the following functions represent a change in the period from the function y = \sin x:
y = \sin \left( 5 x\right)
y = \sin \left( x - 5 \right)
y = 5 \sin x
y = \sin \left( \dfrac{x}{5} \right)
y = \sin x + 5
For each of the following functions:
State the amplitude of the function.
Describe the transformation required to obtain the graph of y = f\left(x\right) from the graph of y = \sin \left(x\right).
To obtain the graph of y = \tan \left( 2 \left(x - \dfrac{\pi}{4}\right)\right) from y = \tan x, two transformations were applied:
Horizontal translation of \dfrac{\pi}{4} units to the right.
Horizontal dilation by a scale factor of \dfrac{1}{2}.
State the transformation that was applied first.
Sketch the graph of each of the following functions for -\pi \leq x \leq \pi:
y = \tan 2 x
y = \tan \left( 2 \left(x - \dfrac{\pi}{4}\right)\right)
Consider the function f\left(x\right) = \cos \left( 2 x + \dfrac{\pi}{3}\right).
Describe the series of transformations required to obtain the graph of f\left(x\right) from the graph of y=\cos x given that a horizontal dilation was performed first.
Sketch the graph of y = f \left( x\right) for 0 \leq x \leq 2\pi.
Describe the series of transformations required to obtain the graph of y = \tan \left( 2 x - \dfrac{\pi}{4}\right) from y = \tan x given that a horizontal dilation was performed first.
Consider the function y = \cos \left( 3 x + \pi\right).
Find the period of the function.
Describe the series of transformations required to obtain the graph of y=f\left(x\right) from the graph of y=\cos x given that a horizontal dilation was performed first.
Sketch a graph of the function for -\pi \leq x \leq \pi.
For each of the following functions:
Find the period.
State the amplitude.
Find the maximum value.
Find the minimum value.
Graph the function for 0 \leq x \leq 2\pi.
Consider the function y = - 4 \tan \dfrac{1}{5} \left(x + \dfrac{\pi}{4}\right).
Find the period of the function.
Describe a series of transformations to obtain the graph of y=f\left(x\right) from the graph of y=\tan x.
Find the range of the function.