Which of the following is the correct domain of y = \tan x?
All real numbers, except integer multiples of 90 \degree.
All real numbers, except odd integer multiples of 90 \degree.
All real numbers, except even integer multiples of 90 \degree.
All real numbers.
Consider the graph of y = \tan x for - 2 \pi \leq x \leq 2 \pi.
State the y-intercept of the graph.
State the period of the function.
State the equations of the vertical asymptotes on the domain 0 \leq x \leq 2\pi.
Does the graph of y=\tan x increase or decrease between any two successive vertical asymptotes?
If x \gt 0, find the least value of x for which \tan x = 0.
Consider the graph of y = \tan x for - 2 \pi \leq x \leq 2 \pi.
Select the word that best describes the graph:
Periodic
Decreasing
Even
Linear
Determine the range of y = \tan x.
As x increases, determine the equation of the next asymptote of the graph after x = \dfrac{7 \pi}{2}.
Consider the graph of y = \tan x for - 2 \pi \leq x \leq 2 \pi.
Determine the sign of \tan x for \\ \pi \leq x < \dfrac{3 \pi}{2}.
Determine the sign of \tan x for \\- \dfrac{\pi}{2} < x \leq 0.
Describe the function y = \tan x as odd, even or neither.
Consider the right triangle containing angle \theta and the graph of y=\cos \theta.
Hence explain what happens to the value of \tan \theta as angle \theta increases from 0 to \dfrac{\pi}{2}, given that \tan \theta is defined as \dfrac{\text{opposite }}{\text{adjacent }}.
Express \tan \theta in terms of \sin \theta and \cos \theta.
Determine the values of \theta for which \cos \theta = 0, given - 2 \pi \leq \theta \leq 2 \pi.
Hence, state the values of \theta between - 2 \pi and 2 \pi for which \tan \theta is undefined.
Complete the table below.
| \theta | -2\pi | -\dfrac{7\pi}{4} | -\dfrac{5\pi}{4} | -\pi | -\dfrac{3\pi}{4} | -\dfrac{\pi}{4} |
|---|---|---|---|---|---|---|
| \tan \theta |
| \theta | 0 | \dfrac{\pi}{4} | \dfrac{3\pi}{4} | \pi | \dfrac{5\pi}{4} | \dfrac{7\pi}{4} | 2\pi |
|---|---|---|---|---|---|---|---|
| \tan \theta |
Hence sketch the graph of y = \tan \theta on the domain - 2 \pi \leq \theta \leq 2 \pi.
How has the graph y = 3 \tan x been transformed from y = \tan x?
Consider the graph of y = a \tan x.
From the graph, determine the value of y when x=\dfrac{\pi}{4}.
If y=\tan x, determine the value of y when x=\dfrac{\pi}{4}.
Find the vertical dilation factor that must be applied to y = \tan x to obtain this graph.
Hence state the value of a.
Determine the equation for each of the following functions, given the equation is in the form y = a \tan x:
On the same set of axes, sketch the graphs of y = \tan x and y = \dfrac{1}{2} \tan x, on the domain -2\pi \leq x \leq 2\pi.
On the same set of axes, sketch the graphs of y = 5 \tan x and y = - 4 \tan x, on the domain -\pi \leq x \leq \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 functions of the form y=\tan bx.
Complete the table identifying the period of the function when b = 1, 2, 3, 4.
State the period of y = \tan b x.
As the value of b increases, describe the effect on the period of y=\tan b x.
| Function | Period |
|---|---|
| \tan x | \pi |
| \tan 2x | |
| \tan 3x | |
| \tan 4x |
Consider the function y = \tan 3x.
Complete the tables of values:
| x | -\dfrac{2\pi}{3} | -\dfrac{7\pi}{12} | -\dfrac{\pi}{2} | -\dfrac{5\pi}{12} | -\dfrac{\pi}{3} | -\dfrac{\pi}{4} | -\dfrac{\pi}{6} | -\dfrac{\pi}{12} |
|---|---|---|---|---|---|---|---|---|
| \tan 3x |
| x | 0 | \dfrac{\pi}{12} | \dfrac{\pi}{6} | \dfrac{\pi}{4} | \dfrac{\pi}{3} | \dfrac{5\pi}{12} | \dfrac{\pi}{2} | \dfrac{7\pi}{12} | \dfrac{2\pi}{3} |
|---|---|---|---|---|---|---|---|---|---|
| \tan 3x |
State the equations of the vertical asymptotes on the domain -\pi \leq x \leq \pi.
Find the interval between the asymptotes of y = \tan 3 x.
Hence, determine the period of y = \tan 3 x.
Write an expression for the period of y = \tan n x.
Sketch the graph of the function y = \tan 3 x on the domain -\pi \leq x \leq \pi.
Consider the function y = \tan 2 x.
Complete the table of values:
| x | -\pi | -\dfrac{3\pi}{4} | -\dfrac{\pi}{4} | 0 | \dfrac{\pi}{4} | \dfrac{3\pi}{4} | \pi |
|---|---|---|---|---|---|---|---|
| 2x | - \dfrac{3\pi}{2} | ||||||
| \tan 2x | \text{Und} |
Find the interval between the asymptotes of y = \tan 2 x.
Hence, determine the period of y = \tan 2 x.
Sketch the graph of y = \tan 2 x on the domain - \pi \leq x \leq \pi.
The graph of f \left( x \right) = \tan x is shown. On the same set of axes, plot g \left( x \right) = \tan \left( \dfrac{1}{2} x\right).
Consider the graph of a function in the form f \left( x \right) = \tan b x:
State the period of the function.
Hence, state the equation of the function.
Consider the function f \left( x \right) = \tan 7 x.
Find the period of the function.
Find the equation of the first four asymptotes to the right of the origin.
If an asymptote of the function in the form g \left( x \right) = \tan b x is known to be x = \dfrac{\pi}{8}, find the equation of g \left( x \right).
Consider the graph of a function in the form y = \tan b x.
State the period of the function.
Hence, 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) displayed on the same coordinate axes:
Which is greater: \alpha or \beta? Explain your answer.
On the same set of axes, sketch the graphs of the functions f \left( x \right) = \tan \dfrac{1}{4} x and \\g \left( x \right) = \tan 4 x, on the domain - 2\pi \leq x \leq 2\pi.
The function f \left( x \right) has the form f \left( x \right) = \tan b x. If two neighbouring asymptotes of this function are known to have equations x = \dfrac{\pi}{12} and x = \dfrac{\pi}{4}, find the exact value of the \\x-intercept between the asymptotes.
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}.
Find the equation of the asymptote lying between the two x-intercepts.
Find the period of the function.
Determine the equation of the function.
The function f \left( x \right) = \tan 6 x is to be graphed on the interval \left[\dfrac{\pi}{12}, \dfrac{5 \pi}{12}\right].
Find the period of the function f \left( x \right) = \tan 6 x.
Find the equations of the asymptotes of the function that occur on this interval.
Find the x-intercepts of the function that occur on this interval.
Hence sketch the function f \left( x \right) = \tan 6 x on the given interval.
Consider the graph of f \left( x \right) = \tan x and three points A\left(0, 0\right), B\left(\dfrac{\pi}{4}, 1\right) and C\left(\dfrac{\pi}{2}, 0\right).
If f \left( x \right) undergoes a transformation to g \left( x \right) = \tan \left(x - \dfrac{\pi}{3}\right), state the coordinates of the following points after the transformation:
A
B
C
Describe the transformation of f \left( x \right) to g \left( x \right).
Hence, sketch the graph of g \left( x \right) = \tan \left(x - \dfrac{\pi}{3}\right) on the domain - \pi \leq x \leq \pi.
Consider the graph shown:
Determine whether the following could be the equation of the graph:
Hence, which of the following statements is true?
The graph of y = \tan \left(x - h\right) is the same as the graph of y = \tan \left(x - h - 180 n\right), for any integer n.
The graph of y = \tan \left(x - h\right) can be reflected in the y-axis to make the graph of y = \tan \left(x - h - 180 n\right), for any integer n.
The graph of y = \tan \left(x - h\right) is different for every value of h.
The graph of y = \tan \left(x - h\right) is the same as the graph of y = \tan \left(x - h - 180 n\right) only when n = \pm 1.
Let f \left( x \right) = \tan x and g \left( x \right) = \tan \left(x - 45 \degree\right).
What can be said of the domain and range of both functions? Explain your answer.
Consider the function y = \tan \left(x - 90 \degree\right).
Does the function have y-intercept?
Determine the period of the function.
How far apart are 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 the graph of the function.
Consider the function y = \tan \left(x - 60 \degree\right).
Determine the value of y at the y-intercept.
Determine the period of the function.
How far apart are 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 the graph the function.
Consider the function y = \tan \left(x - \dfrac{\pi}{4}\right).
Complete the table with exact values for \tan \left(x - \dfrac{\pi}{4}\right):
| x | 0 | \dfrac{\pi}{4} | \dfrac{5\pi}{12} | \dfrac{\pi}{2} | \dfrac{7\pi}{12} | \dfrac{11\pi}{12} | \pi | \dfrac{13\pi}{12} | \dfrac{5\pi}{4} |
|---|---|---|---|---|---|---|---|---|---|
| \tan \left( x - \dfrac{\pi}{4} \right) |
Sketch the graph of y = \tan \left(x - \dfrac{\pi}{4}\right) on the domain - 2\pi \leq x \leq 2\pi.
Describe the transformation that turns the graph of y = \tan x into the graph of \\y = \tan \left(x - h\right).
Consider the graph of a function in the form f \left( x \right) = \tan \left(x - h\right), where 0 \leq h < \pi.
If g\left(x\right) = \tan x, describe the transformation of g \left( x \right) to f \left( x \right).
State the equation of f \left( x \right).
Consider the function f \left( x \right) = \tan \left(x - \dfrac{\pi}{7}\right).
If g\left(x\right) = \tan x, describe the transformation of g \left( x \right) to f \left( x \right).
State the equations of the first four asymptotes of f \left( x \right) to the right of the origin.
On the same set of axes, sketch the graphs of f \left( x \right) = \tan x and g \left( x \right) = \tan \left(x - \dfrac{\pi}{2}\right) on the domain -2\pi \leq x \leq 2\pi.
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 equations of the asymptotes of the function that occur on this interval.
Find the x-intercepts of the function that occur on this interval.
Sketch the function f \left( x \right) = \tan \left(x - \dfrac{\pi}{6}\right) on the interval .
Consider the following functions of the form f\left(x\right)=\tan \left( x-h\right):
Sketch a graph of each of the functions on the domain -\pi \leq x \leq \pi.
p \left( x \right) = \tan \left(x - \dfrac{2 \pi}{3}\right)
r \left( x \right) = \tan \left(x + \dfrac{4\pi}{3}\right)
s \left( x \right) = \tan \left(x + \dfrac{\pi}{3}\right)
r \left( x \right) = \tan \left(x - \dfrac{53 \pi}{3}\right)
Compare the graphs in part (a) and explain your answer.
Consider the graph of f \left( x \right) = \tan \left(x - \beta\right), where 0 \leq \beta < \pi, and g \left( x \right) = \tan \left(x - \alpha\right), where 0 \leq \alpha < \pi.
Determine whether the following statements are true or false:
\beta must be greater than \alpha as the function f(x) has a greater horizontal translation from the graph of y = \tan x.
\alpha must be greater than \beta as the function f(x) has a greater horizontal translation from the graph of y = \tan x.
Consider the graphs of the following functions and state whether or not the graph is the same as y = - \tan x:
y = - \tan \left(x + \dfrac{3 \pi}{4}\right)
y = - \tan \left(x + \pi\right)
y = - \tan \left(x + \dfrac{\pi}{2}\right)
y = - \tan \left(x + 2 \pi\right)
How has the graph y = \tan x + 4 been transformed from y = \tan x?
Determine whether the following statements regarding the graph of y = \tan x, are true or false:
Altering the period will alter the position of the vertical asymptotes.
A phase shift has the same effect as a horizontal translation.
Consider the graph of the given \tan function.
The given graph is the result of what transformation from y = \tan x?
State the equation of the graphed function.
The graph of y = \tan x is shown. Find the equation of the new graph after the following transformations:
Reflected over the y-axis and then translated vertically 3 units down.
Dilated vertically by a scale factor of 2 and translated horizontally \dfrac{\pi}{3} units to the left.
Translated horizontally by \dfrac{\pi}{3} units right and then dilated horizontally by a scale factor of \dfrac{1}{2}.
Sketch the graph of the following functions:
y = \tan \left( 2 \left(x - 45\right)\right)
y = \tan \left( 2 x + 90 \degree\right)
Sketch the graph of the function y = \tan \left( 2 \left(x - \dfrac{\pi}{4}\right)\right).
Consider the function y = \tan \left( 2 x + \dfrac{\pi}{3}\right):
Rewrite the function so that it is in the form y = \tan \left( a \left(x + b\right)\right).
Hence, sketch the graph of y = \tan \left( 2 \left(x + \dfrac{\pi}{6}\right)\right).
The graph y = \tan x, as shown, is reflected over the y-axis and then translated vertically 3 units down.
Write the equation of the new graph.
Consider the graph of the given \tan function.
The given graph is the result of what transformation from y = \tan x?
State the equation of the graphed function.
For each of the following functions:
Determine the y-intercept.
Determine the interval between the vertical asymptotes of the function.
Hence, state the period 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 the graph the function on the domain -\pi \leq x \leq \pi.
y = - \tan x
y = \tan \left(x + \dfrac{\pi}{3}\right)
y = \tan \left(\dfrac{x}{2}\right)
For each of the following functions:
Find the value of y when x = \dfrac{\pi}{4}.
Determine the period of the function.
Hence, state the interval 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 the graph the function on the domain -\pi \leq x \leq \pi.
y = 5 \tan x + 3
y = 4 \tan 3 x
Describe how the graph of each of the following functions has been transformed from the function y = \tan x:
y = - 5 \tan x
y = 3\tan x + 2
y = \tan \left(3\left(x + \dfrac{\pi}{4}\right)\right)
y = \tan \left( 2 x - \dfrac{\pi}{4}\right)
Determine the following features for each of the given functions:
Period
Phase shift
Range
Midline
y = 6 - 3 \tan \left(x + \dfrac{\pi}{3}\right)
y = - 4 \tan \left(\dfrac{1}{5}x + \dfrac{\pi}{20}\right)