9. Graphs of Trigonometric Functions

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`|<