Lesson

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

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

The general form of the tan functions is

$f\left(x\right)=a\tan\left(bx-c\right)+d$`f`(`x`)=`a``t``a``n`(`b``x`−`c`)+`d`

or

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

Here is a summary of our transformations for $y=\tan x$`y`=`t``a``n``x`:

**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.

**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 $\frac{c}{b}$
`c``b` from the equation

How has the graph $y=3\tan x$`y`=3`t``a``n``x` been transformed from $y=\tan x$`y`=`t``a``n``x`?

Choose one of the following options:

Vertical dilation by a scale factor of $3$3.

AHorizontal dilation by a scale factor of $3$3.

BHorizontal translation by $3$3 to the right.

CVertical dilation by a scale factor of $\frac{1}{3}$13.

DVertical dilation by a scale factor of $3$3.

AHorizontal dilation by a scale factor of $3$3.

BHorizontal translation by $3$3 to the right.

CVertical dilation by a scale factor of $\frac{1}{3}$13.

D

Select all functions that have the same graph as $y=-\tan x$`y`=−`t``a``n``x`.

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

`y`=−`t``a``n`(`x`+135°)A$y=-\tan\left(x+180^\circ\right)$

`y`=−`t``a``n`(`x`+180°)B$y=-\tan\left(x-360^\circ\right)$

`y`=−`t``a``n`(`x`−360°)C$y=-\tan\left(x+90^\circ\right)$

`y`=−`t``a``n`(`x`+90°)D$y=-\tan\left(x+135^\circ\right)$

`y`=−`t``a``n`(`x`+135°)A$y=-\tan\left(x+180^\circ\right)$

`y`=−`t``a``n`(`x`+180°)B$y=-\tan\left(x-360^\circ\right)$

`y`=−`t``a``n`(`x`−360°)C$y=-\tan\left(x+90^\circ\right)$

`y`=−`t``a``n`(`x`+90°)D

Consider the function $y=\tan4x-3$`y`=`t``a``n`4`x`−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...

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.

Consider the graph of $y=\tan x$`y`=`t``a``n``x` below.

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

All real values of $x$