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y = cos ( k θ ) y i n t = cos ( 0 ) = 1

Functions of the form y = tan ( k θ )

In the equation, y = tan ( k θ ) , k is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function f ( θ ) = tan ( 2 θ ) .

The graph of tan ( 2 θ ) (solid line) and the graph of g ( θ ) = tan ( θ ) (dotted line). The asymptotes are shown as dashed lines.

Functions of the form y = tan ( k θ )

On the same set of axes, plot the following graphs:

  1. a ( θ ) = tan 0 , 5 θ
  2. b ( θ ) = tan 1 θ
  3. c ( θ ) = tan 1 , 5 θ
  4. d ( θ ) = tan 2 θ
  5. e ( θ ) = tan 2 , 5 θ

Use your results to deduce the effect of k .

You should have found that, once again, the value of k affects the periodicity (i.e. frequency) of the graph. As k increases, the graph is more tightly packed. As k decreases, the graph is more spread out. The period of the tan graph is given by 180 k .

These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = tan ( k θ ) .
k > 0 k < 0

Domain and range

For f ( θ ) = tan ( k θ ) , the domain of one branch is { θ : θ ( - 90 k , 90 k ) } because the function is undefined for θ = - 90 k and θ = 90 k .

The range of f ( θ ) = tan ( k θ ) is { f ( θ ) : f ( θ ) ( - , ) } .

Intercepts

For functions of the form, y = tan ( k θ ) , the details of calculating the intercepts with the x and y axis are given.

There are many x -intercepts; each one is halfway between the asymptotes.

The y -intercept is calculated as follows:

y = tan ( k θ ) y i n t = tan ( 0 ) = 0

Asymptotes

The graph of tan k θ has asymptotes because as k θ approaches 90 , tan k θ approaches infinity. In other words, there is no defined value of the function at the asymptote values.

Functions of the form y = sin ( θ + p )

In the equation, y = sin ( θ + p ) , p is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function f ( θ ) = sin ( θ + 30 ) .

Graph of f ( θ ) = sin ( θ + 30 ) (solid line) and the graph of g ( θ ) = sin ( θ ) (dotted line).

Functions of the form y = sin ( θ + p )

On the same set of axes, plot the following graphs:

  1. a ( θ ) = sin ( θ - 90 )
  2. b ( θ ) = sin ( θ - 60 )
  3. c ( θ ) = sin θ
  4. d ( θ ) = sin ( θ + 90 )
  5. e ( θ ) = sin ( θ + 180 )

Use your results to deduce the effect of p .

You should have found that the value of p affects the position of the graph along the y -axis (i.e. the y -intercept) and the position of the graph along the x -axis (i.e. the phase shift ). The p value shifts the graph horizontally. If p is positive, the graph shifts left and if p is negative tha graph shifts right.

These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = sin ( θ + p ) . The curve y = sin ( θ ) is plotted with a dotted line.
p > 0 p < 0

Domain and range

For f ( θ ) = sin ( θ + p ) , the domain is { θ : θ R } because there is no value of θ R for which f ( θ ) is undefined.

The range of f ( θ ) = sin ( θ + p ) is { f ( θ ) : f ( θ ) [ - 1 , 1 ] } .

Intercepts

For functions of the form, y = sin ( θ + p ) , the details of calculating the intercept with the y axis are given.

The y -intercept is calculated as follows: set θ = 0

y = sin ( θ + p ) y i n t = sin ( 0 + p ) = sin ( p )

Functions of the form y = cos ( θ + p )

In the equation, y = cos ( θ + p ) , p is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function f ( θ ) = cos ( θ + 30 ) .

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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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