# 9.03 Graphing tangent functions

Lesson

## Transformations of tangent curves and equations

We already know that transformations to curves, graphs or equations mean that we are doing one of four things:

• horizontal translation - shifting the graph horizontally (phase shift)
• vertical translation - shifting the curve vertically
• reflection - reflecting the curve in the $y$y-axis
• dilation - changing the dilation of the curve

#### Exploration

Use the geogebra applet below to adjust the constants in $y=a\tan b\left(x-c\right)+d$y=atanb(xc)+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?

### 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(bxc)+d

or

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

1. 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

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.

1. $y=-\tan\left(x+135^\circ\right)$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(x360°)

C

$y=-\tan\left(x+90^\circ\right)$y=tan(x+90°)

D

$y=-\tan\left(x+135^\circ\right)$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(x360°)

C

$y=-\tan\left(x+90^\circ\right)$y=tan(x+90°)

D

##### Question 3

Consider the function $y=\tan4x-3$y=tan4x3.

1. Determine the $y$y-intercept.

2. Determine the period of the function.

3. How far apart are the asymptotes of the function?

4. State the first asymptote of the function for $x\ge0$x0.

5. State the first asymptote of the function for $x\le0$x0.

6. Graph the function.

## 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 $x$