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.
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{Undefined} |
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.
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 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.
Which is greater in value: \alpha or \beta? Explain your answer.
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)
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.
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}.
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)
y = \tan \left( 3 \left(x + \dfrac{\pi}{4}\right)\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)