Use the geogebra applet below to adjust the constants in $y=a\tan b\left(x-c\right)+d$y=atanb(x−c)+d and observe how it affects the graph. Try to answer the following questions.

Which constants affect the position of the vertical asymptotes? Which ones don't?
Which constants translate the graph, leaving the shape unchanged? Which ones affect the size?
Which constants change the period of the graph? Which ones don't?
Do any of these constants affect the range of the graph? If so, which ones?

Created with Geogebra

The general form

The general form of the tan functions is

$f\left(x\right)=a\tan\left(bx-c\right)+d$f(x)=atan(bx−c)+d

$f\left(x\right)=a\tan b\left(x-\frac{c}{b}\right)+d$f(x)=atanb(x−cb)+d

Here is a summary of our transformations for $y=\tan x$y=tanx:

Dilations:

The vertical dilation (a stretching or shrinking in the same direction as the $y$y-axis) occurs when the value of a is not one.
If $|a|>1$|a|>1, then the graph is stretched
If $|a|<1$|a|<1 then the graph is compressed
Have another look at the applet above now, and change the a value. Can you see the stretching and shrinking?
The horizontal dilation (a stretching of shrinking in the same direction as the $x$x-axis) occurs when the period is changed, see the next point.

Dilations of the tangent function.

Reflection:

If $a$a is negative, then there is a reflection. Have a look at the applet above and make $a$a negative, can you see what this does to the curve?

Period:

The period is calculated using $\frac{\pi}{|b|}$π|b| for radians or $\frac{180^\circ}{\left|b\right|}$180°|b|
From a graph you can read the value for the period directly by measuring the distance for one complete cycle.
For a tangent function this occurs between the asymptotes.
When the period is increased, then the graphs horizontal dilation can be described as a stretch.
When the period is decrease, then the graphs horizontal dilation can be described as being shrunk or compressed.

Vertical translations:

The whole function is shifted by $d$d units.
From an equation you can read the value from the equation directly. If $d>0$d>0 then the graph is translated up, if $d<0$d<0 then the graph is translated down.

Phase shift:

The phase shift is the comparable transformation to a horizontal translation. The phase shift is found by calculating $\frac{c}{b}$cb from the equation

A phase shift of the tangent function.

Practice questions

Question 1

How has the graph $y=3\tan x$y=3tanx been transformed from $y=\tan x$y=tanx?

Choose one of the following options:
Vertical dilation by a scale factor of $3$3.
A
Horizontal dilation by a scale factor of $3$3.
B
Horizontal translation by $3$3 to the right.
C
Vertical dilation by a scale factor of $\frac{1}{3}$13.
D

Question 2

Select all functions that have the same graph as $y=-\tan x$y=−tanx.

$y=-\tan\left(x+135^\circ\right)$y=−tan(x+135°)
A
$y=-\tan\left(x+180^\circ\right)$y=−tan(x+180°)
B
$y=-\tan\left(x-360^\circ\right)$y=−tan(x−360°)
C
$y=-\tan\left(x+90^\circ\right)$y=−tan(x+90°)
D

Question 3

Consider the function $y=\tan4x-3$y=tan4x−3.

Answer the following questions in radians, where appropriate.

Determine the $y$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 $x\ge0$x≥0.
State the first asymptote of the function for $x\le0$x≤0.
Graph the function.

Loading Graph...

Domain and range of tangent curves

The domain of a function is the set of all values that the independent variable (usually $x$x) can take and the range of a function is the set of all values that the dependent variable (usually $y$y) can attain.

Graphically speaking, we can determine the domain by observing the values of $x$x for which the function is defined over. We can also determine the range by observing the heights of each point on the graph.

Exploration

Consider the graph of $y=\tan x$y=tanx below.

$y=\tan x$y=tanx

Domain

Notice that the graph of $y=\tan x$y=tanx is undefined at periodic intervals of length $\pi$π or $180^\circ$180°. We state the domain as being:

All real values of $x$x, except where $x=k\times180^\circ+90^\circ$x=k×180°+90° for any integer $k$k.

All real values of $x$x, except where $x=k\times\pi+\frac{\pi}{2}$x=k×π+π2 for any integer $k$k.

More generally speaking, we can consider the function $y=a\tan\left(bx-c\right)+d$y=atan(bx−c)+d where $a,b,c,d$a,b,c,d are constants. The horizontal transformations $b$b and $c$c will impact the domain.

Conceptually, the domain is all real values of $x$x, except for the asymptotes.

Careful!

Notice that the constants $a$a and $d$d do not affect the domain of the function. This is because $a$a and $d$d relate to a vertical dilation and translation, which does not change the position of the vertical asymptotes.

Range

The graph of $y=\tan x$y=tanx has no minimums, maximums, horizontal asymptotes or holes. In other words, the range is all real values of $y$y or $\left(-\infty,\infty\right)$(−∞,∞). The only exception is when we restrict the graph to a smaller domain as shown below.

$y=\tan x$y=tanx over the domain $\left[0^\circ,90^\circ\right)$[0°,90°)

In this case, the range is no longer all real values of $y$y but instead, $\left[0,\infty\right)$[0,∞). We include $y=0$y=0 in the range because the domain includes the value $x=0^\circ$x=0°.

Remember!

For a function of the form $y=a\tan\left(bx-c\right)+d$y=atan(bx−c)+d, where $a,b,c,d$a,b,c,d are constants:

The domain is all real $x$x, except where $x=\frac{k\times180^\circ}{b}+\frac{90^\circ+c}{b}$x=k×180°b+90°+cb for any integer $k$k
The range is $\left(-\infty,\infty\right)$(−∞,∞)

However on a restricted domain, the range can be any subset of the real numbers.

Practice questions

Question 4

Let $f\left(x\right)=\tan x$f(x)=tanx and $g\left(x\right)=\tan2x$g(x)=tan2x.

Select the correct statement from the options below.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same range, but different domains.
A
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have different domains and ranges.
B
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain, but different ranges.
C
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain and range.
D
Select the most appropriate explanation for your answer to part (a).
Multiplying $x$x by $2$2 translates a function horizontally.
A
Multiplying $x$x by $2$2 translates a function vertically.
B
Multiplying $x$x by $2$2 does not change the position of the asymptotes, and a vertical shift will not affect a range of $\left(-\infty,\infty\right)$(−∞,∞).
C
Multiplying $x$x by $2$2 changes the position of the asymptotes.
D

question 5

The graph of $f\left(x\right)=\tan\left(x-\frac{\pi}{3}\right)$f(x)=tan(x−π3) is shown.

Loading Graph...

Select the correct domain of $f\left(x\right)$f(x).
All real $x$x except when $x=\pi k+\frac{5\pi}{6}$x=πk+5π6 for all integer $k$k.
A
All real $x$x.
B
All real $x$x except when $x=\pi k+\frac{\pi}{2}$x=πk+π2 for all integer $k$k.
C
All real $x$x except when $x=\pi k+\frac{\pi}{6}$x=πk+π6 for all integer $k$k.
D
State the range of $f\left(x\right)$f(x) using interval notation.

Range: $\left(\editable{},\editable{}\right)$(,)

question 6

Let $f\left(x\right)=\tan x$f(x)=tanx and $g\left(x\right)=\tan2x$g(x)=tan2x.

Select the correct statement from the options below.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain and range.
A
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same range, but different domains.
B
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have different domains and ranges.
C
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain, but different ranges.
D
Select the most appropriate explanation for your answer to part (a).
Multiplying $x$x by $2$2 does not change the position of the asymptotes, and a vertical shift will not affect a range of $\left(-\infty,\infty\right)$(−∞,∞).
A
Multiplying $x$x by $2$2 translates a function vertically.
B
Multiplying $x$x by $2$2 changes the position of the asymptotes.
C
Multiplying $x$x by $2$2 translates a function horizontally.
D

Outcomes

F-TF.5

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

9.03 Graphing tangent functions

Transformations of tangent curves and equations

Exploration

The general form

Practice questions

Question 1

Question 2

Question 3

Domain and range of tangent curves

Exploration

Domain

Range

Practice questions

Question 4

question 5

question 6

Outcomes

F-TF.5

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