Trigonometric Functions - Ariel and Gabrielle


Formulas:
Y = A tan (Bx-C)
Y = A cot (Bx-C)

A = Amplitude
*Amplitude is undefined (has no maximum or minimum point)

|A| > 1 steeper graph
|A|< 1 flatter graph
*When A is negative the graph is reflected over the x-axis


Period = π/B
*Note that this period is different from sine and cosine graphs which have periods that are defined by the formula: 2π/B

Phase Shift = C/B
*The direction of the phase shift is determined by whether the product is negative or positive. If C/B is positive, the graph shifts to the LEFT. If C/B is negative, the graph shifts to the RIGHT.

FOR TANGENT GRAPHS WITHOUT A PHASE SHIFT, THE ASYMPTOTES CENTER THE Y-AXIS.
FOR COTANGENT GRAPHS WITHOUT A PHASE SHIFT, THE ASYMPTOTES BEGIN ON Y-AXIS.

Examples:

Y = tan (2x)
  • Graph the tangent function according to its period. (Formula: π/B)
  • Draw the asymptotes. (Tangent graph has asymptotes that center the y-axis)
  • Note that tangent graphs intersect the origin
  • Tangent graphs move up to the right and down to the left

SCAN0002.JPG



Y = tan (x + π/2)
  • Graph the tangent function according to its period. (Formula: π/B) Graph the function without the phase shift first.
  • Draw asymptotes according to the graph without the phase shift (Centering the y-axis since it is a tangent function
  • Shift the graph and its asymptotes C/B units. (Since C/B is positive, the graph shifts to the left π/2 units)

SCAN0003.JPG



Y = cot (x/2)
  • Graph the tangent function according to its period (Formula: π/B)
  • Draw the asymptotes. (Cotangent graphs have asymptotes that begin on the y-axis.
  • Cotangent graphs move up to the left and down to the right

SCAN0004.JPG



Y = cot (x + π/2)
  • Graph the tangent function according to its period (Formula: π/B). Graph the function without the phase shift first.
  • Draw asymptotes according to the graph without the phase shift. (Begin on the origin and draw asymptotes to the left and right)
  • Shift the graph and its asymptotes C/B units. (Since C/B is positive, the graph shifts to the left π/2 units.

SCAN0005.JPG



NEGATIVE COTANGENT GRAPH FOLLOWS THE SAME RULES AS A NEGATIVE TANGENT GRAPH!
--First draw the same function except positively then flip the function.
math2.jpg




NOTE:
If it is a negative cotangent function, the graph will have a tangent shape (down the left and up to the right). If it is a negative tangent function, the graph will have a cotangent shape (up the left and down to the right).




This example includes all of the shifts, period changes, and amplitude changes. If you can master this, you have mastered tangent and cotangent graphs!

math_big_.jpg



HERE ARE SOME EXTRA PROBLEMS TO HELP YOU MASTER TANGENT AND COTANGENT! COPY AND PASTE INTO A WORD DOC!

Level 1 Trigonometry / Pre-calculus Name: _

Here are extra problems that will help you master cotangent and tangent graphs. Be sure to show two full periods, label the axes, and draw all asymptotes! (Hint: determine the period, asymptotes, amplitude, and phase shift prior to drawing the graph. This will make it easier to draw the graph later on. Also, drawing each isolated shift from the parent graph will make the graph neater and easier to interpret.)


  1. f(x) = cot (2x)






  1. f(x) = -cot (x/2)







  1. f(x) = cot (Πx/4)








  1. f(x) = cot (x – Π/2)







  1. f(x) = -cot (Πx/2 + Π/4)






  1. f(x) = tan (x/2)







  1. f(x) = -tan (Πx/4)








  1. f(x) = tan (x + Π/4)








  1. f(x) = tan (4x – Π/2)






  1. f(x) = -tan (Πx/4 – Π/2)








GRAPHS OF TANGENT AND COTANGENT
BY DAVID AND SEAN
The formula for tangent and cotangent are as follows:
For tangent- a•tan(bx+c)+d
For cotangent- a•cot(bx+c)+d


a is the amplitude of the graph, but because it has no maximum and the graph would only become steeper by a miniscule amount it does not need to be considered when graphing a tangent or contangent function except when it is a negative number. If it is negative the graph is reflected over the x-axis.

b is used to find the period of the function and the phase shift.

c is also used to find the phase shift.

d is when there is a vertical shift. When it is positive the graph moves up and when it is negative the graph moves down.

•The equation to find the period is π/b.

•The equation to find the phase shift is c/b. If the answer is positive the phase shift is to the left and if it is negative then the pahse shift is to the right. If there is NO phase shift than the tangent graph crosses the x-axis at the origin while the cotangent graph contains an asymptote at the origin.

A tangent graph has asymptotes that are a distance of Pi apart on the graph starting at π/2.
A cotangent graph has asymptotes that are a distance of Pi apart on the graph but the asymptotes start at the origin.

A regular tangent graph looks like the following:

tanx.gif

A regular cotangent graph looks like the following:

cotx.gif

Try these next problems and scroll down to see the answers.

  1. tan(4x)
  2. -3•tan(6x)+2
  3. 2•tan(3x+π/2)
  4. 4•tan(4x+3)+7
  5. cot(4x)+3
  6. -8•cot(7x+3)
  7. cot(2x-π/3)-4









Answers:

tan(4x)sean3.JPG

-3•tan(6x)+2

tangraph.JPG

2•tan(3x+π/2)
tangraph2.JPG
4•tan(4x+3)+7
tangraph3.JPG

cot(4x)+3

cot.JPG

-8•cot(7x+3)
cot2.JPG

cot(2x-π/3)-4
cot3.JPG

Still need more help?
Visit this page for tangent help.

See how changing a b c or d in a tangent function changes the graph. Click Here.
See the same thing for cotangents. Click Here.

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