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Algebra 2 - Grades 9-12

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|<