United States of America

# 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|<