 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|<