Consider the equation y = \sin x.
Complete the table with values in exact form:
| x | 0 | \dfrac{\pi}{6} | \dfrac{\pi}{2} | \dfrac{5 \pi}{6} | \pi | \dfrac{7 \pi}{6} | \dfrac{3 \pi}{2} | \dfrac{11 \pi}{6} | 2 \pi |
|---|---|---|---|---|---|---|---|---|---|
| \sin x |
Sketch a graph for y = \sin x on the domain -2\pi \leq 0 \leq 2\pi.
State the value of \sin \left(-2 \pi\right).
State the sign of \sin \left( \dfrac{- \pi}{12} \right).
State the sign of \sin \dfrac{13 \pi}{12}.
Which quadrant of a unit circle does an angle with measure \dfrac{13 \pi}{12} lie in?
Consider the equation y = \cos x.
Complete the table with values in exact form:
| x | 0 | \dfrac{\pi}{3} | \dfrac{\pi}{2} | \dfrac{2 \pi}{3} | \pi | \dfrac{4 \pi}{3} | \dfrac{3 \pi}{2} | \dfrac{5 \pi}{3} | 2 \pi |
|---|---|---|---|---|---|---|---|---|---|
| \cos x |
Sketch a graph for y = \cos x on the domain -2\pi \leq 0 \leq 2\pi.
State the value of \cos \pi.
State the sign of \cos \left( \dfrac{- \pi}{4} \right).
State the sign of \cos \dfrac{11 \pi}{6}.
Which quadrant of a unit circle does an angle with measure \dfrac{11 \pi}{6} lie in?
Consider the graph of y = \sin x given below:
Using the graph, what is the sign of \sin \dfrac{13 \pi}{12}?
Which quadrant does the angle \dfrac{13 \pi}{12} lie in?
Consider the following unit circle:
State the range of y = \cos x.
State the range of y = \sin x.
How often does the graph of y = \cos x repeat?
How often does the graph of y = \sin x repeat?
Consider the curve y = \sin x drawn below:
If one cycle of the graph of y = \sin x starts at x = 0, when does the next cycle start?
List the regions on the graph that y = \sin x is decreasing.
State the x-intercept in the region 0 < x < 2 \pi.
Consider the curve y = \cos x drawn below:
What are the x-intercepts in the region - 2 \pi < x < 0?
As x approaches infinity, what y-values does the graph of y = \cos x stay between?
List the regions on the graph that y = \cos x is increasing.
Consider the functions y = \sin x and y = \cos x.
State the amplitude of both the graphs of these functions.
State the period of both the graphs of these functions.
Consider the graphs of \\ y = \cos x and y = \cos x + 2 shown:
Describe how to transform the graph of \\ y = \cos x to get y = \cos x + 2.
The function y = \cos x + 5 is translated 4 units up.
Write down the equation of the new function after the translation.
What is the maximum value of the new function?
Which of the following functions has a different amplitude to y = \cos x?
y = \cos 3 x
y = \cos \left( x - 3 \right)
y = \cos x + 3
y = 3 \cos x
Determine the equation of the graphed function given that it is of the form \\ y = a \sin x or y = a \cos x.
Determine the equation of the graphed function given that it is of the form \\ y = \sin b x or y = \cos b x, where b is positive.
Consider the graph of y = \sin x:
At which value of x in the given domain would y = - \sin x have a maximum value?
Consider the function y = - 3 \cos x.
State the maximum value of the function.
State the minimum value of the function.
State the amplitude of the function.
State the two transformations that are required to turn the graph of y = \cos x into the graph of y = - 3 \cos x.
Determine whether f\left(x\right)=\sin 2 x is an odd function, even function, or neither.
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.
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
Consider the function y = - 5 \cos x.
State the amplitude of the function.
Graph the function for 0 \leq x \leq 2\pi.
Sketch the graph of the function y = 2 + \sin x for 0 \leq x \leq 2\pi.
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).
State the period of g \left( x \right).
What transformation of the graph of f \left( x \right) results in the graph of g \left( x \right)?
Graph y = f \left( x \right) and y = g \left( x \right) on the same number plane for 0 \leq x \leq 2\pi.
Is the amplitude of g \left( x \right) different to the amplitude of f \left( x \right)?
Consider the function y = 4 \sin x.
State the amplitude of the function.
Graph the function for 0 \leq x \leq 2\pi.
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).
State the period of g \left( x \right).
What transformation of the graph of f \left( x \right) results in the graph of g \left( x \right)?
Graph y = f(x) and y = g \left( x \right) on the number plane for 0 \leq x \leq 2\pi.
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 |
Complete the following sentence:
The graph of the sine function crosses the x-axis for all numbers of the form ⬚, where n is an integer.
Consider the given graph of \\ y = \cos \left(x + \dfrac{\pi}{2}\right):
State the amplitude of the function.
Describe how the graph of y = \cos x can be transformed into the graph of \\ y = \cos \left(x + \dfrac{\pi}{2}\right).
Determine the equation of the graphed function given that it is of the form \\ y = \sin \left(x - c\right), where c is the least positive value possible.
Determine the values of c in the region - 2 \pi \leq c \leq 2 \pi that make: y = \sin \left(x - c\right) the same as y = \cos x.
What two transformations could be used to turn the graph of y = \cos x into the graph of \\ y = - \cos x + 3?
Describe the three transformations required to turn the graph of y = \cos x into the graph of y = - 5 \cos \left( 4 x\right).
Consider the function y = \cos x and the following graph:
Describe the transformations required to turn the graph of y = \cos x into the given graph.
Write the equation for the given graph?
Consider the function y = 3 \sin \left(\dfrac{x}{2}\right).
Find the period of the function in radians.
Within the domain 0 \leq x \leq 4 \pi, what are the x-intercepts of y = 3 \sin \left(\dfrac{x}{2}\right)?
For 0 \leq x \leq 4 \pi, the function has a maximum value of 3. Determine the value of x at which the maximum value occurs in this domain.
The functions f \left( x \right) and g \left( x \right) = af \left( \dfrac{x}{b} \right) have been graphed as shown:
Describe the transformations that occurred on f \left( x \right) to get g \left( x \right).
Determine the value of a.
Determine the value of b.
The functions f \left( x \right) and \\ g \left( x \right) = f \left( x - c \right) - d have been graphed as shown:
Describe the transformations that occurred on f \left( x \right) to get g \left( x \right).
Determine the value of d.
Determine the smallest positive value of c.
Consider the graphs of y = \cos x and \\ y = 3 \cos \left(x - \dfrac{\pi}{4}\right):
List the type of transformations that have occurred on y = \cos x to get \\ y = 3 \cos \left(x - \dfrac{\pi}{4}\right).
Describe how the amplitude of y = \cos x changed.
What phase shift has y = \cos x undergone to get y = 3 \cos \left(x - \dfrac{\pi}{4}\right)?
Consider the graph of y = \sin x below. Its first maximum point for x \geq 0 is at \left(\dfrac{\pi}{2}, 1\right).
By considering the transformation that has taken place, state the coordinates of the first maximum point of each of the following functions for x \geq 0:
y = 5 \sin x
y = - 5 \sin x
y = \sin x + 2
y = \sin 3 x
y = \sin \left(x - \dfrac{\pi}{4}\right)
y = 5 \sin x + 2
Consider the graph of y = \cos x below. Its first maximum point for x \geq 0 is at \left(0, 1\right).
By considering the transformation that has taken place, state the coordinates of the first maximum point of each of the following functions for x \geq 0:
y = \cos \left(x + \dfrac{\pi}{3}\right)
y = 5 \cos \left(x - \dfrac{\pi}{3}\right)
y = 2 - 5 \cos x
y = \cos \left(\dfrac{x}{4}\right)
y = 5 \cos 4 x - 2
y = \cos \left(x - \dfrac{\pi}{3}\right) + 2
The graph of y = \cos x and another function that is a result of certain transformations on \\ y = \cos x is shown below:
List the type of transformations that have occurred.
Complete the following statement:
The graph of y = \cos x has decreased its period by a factor of ⬚ and then has undergone a phase shift of ⬚ to the left.
Find the equation of the transformed graph.
The graph of y = \cos x undergoes the series of transformations in the following order:
The graph is reflected across the x-axis.
The graph is then horizontally translated to the left by \dfrac{\pi}{6} radians.
The graph is then vertically translated upwards by 5 units.
Find the equation of the transformed graph in the form y = - \cos \left(x + c\right) + d where c is the lowest positive value in radians.
For each of the trigonometric functions below:
State the period in radians.
For each of the following functions:
State the period.
State the amplitude.
Write down the phase shift of the function in radians.
Graph the function for -\pi \leq x \leq \pi.
Consider the functions y = 2 \cos \left(x - \dfrac{\pi}{3}\right) and y = \sin \left(\dfrac{x}{4}\right).
Graph the two functions on the same set of axes for 0 \leq x \leq 2\pi.
Hence, determine the number of solutions of the equation 2 \cos \left(x - \dfrac{\pi}{3}\right) - \sin \left(\dfrac{x}{4}\right) = 0 for 0 \leq x \leq 4 \pi.
Consider the equation y = \tan x.
Complete the table with values in exact form:
| x | 0 | \dfrac{\pi}{4} | \dfrac{\pi}{2} | \dfrac{3 \pi}{4} | \pi | \dfrac{5 \pi}{4} | \dfrac{3 \pi}{2} | \dfrac{7 \pi}{4} | 2 \pi |
|---|---|---|---|---|---|---|---|---|---|
| \tan x |
Sketch the graph of y = \tan x on the domain -2\pi \leq 0 \leq 2\pi.
Graph the line y = 1 on the same coordinate plane.
Hence, state the exact solutions to the equation \tan x = 1 over this domain.
State the value of \tan \left(-2 \pi\right).
State the sign of \tan \left( \dfrac{- \pi}{6} \right).
State the sign of \tan \dfrac{9 \pi}{5}.
Which quadrant of a unit circle does an angle with measure \dfrac{9 \pi}{5} lie in?
Consider the graph of y = \tan x shown:
State the sign of \tan \dfrac{9 \pi}{5}.
Which quadrant does the angle \dfrac{9 \pi}{5} lie in?
Consider the function y = \tan \theta.
\tan \theta is defined as \dfrac{\text{opposite }}{\text{adjacent }} for 0 \leq \theta < \dfrac{\pi}{2} in a right-angled triangle.
What happens to the value of \tan \theta as \theta increases from 0 to \dfrac{\pi}{2}?
The graph of y = \cos x for 0 \leq x \leq 2 \pi is provided. For what values of x is \cos x = 0?
Hence, for what values of x between 0 and 2 \pi is \tan x undefined?
Complete the table below:
| x | 0 | \dfrac{\pi}{4} | \dfrac{3 \pi}{4} | \pi | \dfrac{5 \pi}{4} | \dfrac{7 \pi}{4} | 2 \pi |
|---|---|---|---|---|---|---|---|
| \tan x |
Sketch the graph of y = \tan x for 0 \leq x \leq 2 \pi.
Which of the following terms describes the graph?
Periodic
Decreasing
Even
Linear
Which of the following terms is not an appropriate description of the graph of y = \tan x?
Amplitude
Range
Period
Asymptotes
State the period of y = \tan x in radians.
State the range of y = \tan x.
As x increases, what would be the next asymptote of the graph after x = \dfrac{7 \pi}{2}?
Consider the unit circle shown:
Express \tan \theta in terms of \sin \theta and \cos \theta.
Does the graph of y = \tan x repeat in regular intervals? Explain your answer.
Consider the function f \left( x \right) = \tan x graphed and the function g \left( x \right) = \tan \left(x - \dfrac{\pi}{3}\right).
The point A on the graph of f \left( x \right) has the coordinates \left(0, 0\right).
What are the coordinates of the corresponding point on the graph of g \left( x \right)?
The point B on the graph of f \left( x \right) has the coordinates \left(\dfrac{\pi}{4}, 1\right).
What are the coordinates of the corresponding point on the graph of g \left( x \right)?
The graph of f \left( x \right) has an asymptote passing through point C with coordinates \left(\dfrac{\pi}{2}, 0\right).
What are the coordinates of the corresponding point on the graph of g \left( x \right)?
Hence, apply a phase shift to the graph of f \left( x \right) = \tan x to sketch the graph of \\ g \left( x \right) = \tan \left(x - \dfrac{\pi}{3}\right) for 0 \leq x \leq 2 \pi.
On the same set of axes, sketch the graph of y = \tan x and y = \dfrac{1}{2} \tan x for -2 \pi \leq x \leq 2 \pi.
On the same set of axes, sketch the graphs of the functions f \left( x \right) = - \dfrac{1}{2} \tan x and \\g \left( x \right) = 2 \tan x, on the domain -\pi \leq x \leq \pi.
Consider the function y = - 4 \tan \dfrac{1}{5} \left(x + \dfrac{\pi}{4}\right).
Find the period of the function, giving your answer in radians.
Find the phase shift of the function, giving your answer in radians.
State the range of the function.
Consider the function y = 6 - 3 \tan \left(x + \dfrac{\pi}{3}\right).
Find the period of the function, giving your answer in radians.
Find the phase shift of the function, giving your answer in radians.
State the range of the function.
How has the graph y = \tan \left( 2 x - \dfrac{\pi}{4}\right) been transformed from y = \tan x?
For each of the functions below:
Find the y-intercept.
Find the value of y when x = \dfrac{\pi}{4}.
Find the period of the function.
Find the distance between the asymptotes of the function.
State the equation of the first asymptote of the function for x \geq 0.
State the equation of the first asymptote of the function for x \leq 0.
Sketch a graph of the function for -2 \pi \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 function in radians.
Sketch the graph of the function 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.