Math 30P   Sine/Cosine   Translations    Stretches   Reflections   Inverses   Reciprocals   Combinations   Prerequisites

Sine and Cosine Functions

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If you understand the pattern(s) the concept(s) in What to Learn should make sense.

What to Learn

 

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Standard Form

Parameters:

a

(amplitude) vertical stretch → reflection in x-axis when a < 0 and k = 0
                                           → reflection in y = k when a < 0 and k ≠ 0

  • changes range, but not domain
1/b

horizontal stretch → reflection in y-axis when b < 0 and h = 0
                             → reflection in x = h when b < 0 and h ≠ 0

  • does not change domain or range
  • If f [sin(4x-12)], rewrite as: f [sin(4(x-3))]
  • If f [cos(4x-12)], rewrite as: f [cos(4(x-3))]
h

horizontal phase shift/translation

  • does not change domain or range
k

vertical displacement/translation (midline)

  • changes range, but not domain

(h, k)

Location of red data point on graph

  • is at the origin when (h, k) = (0, 0). At this point (h, k) is the center of the graph.

Note:

Tangent - not shown Sine/Cosine
amplitude - not applicable | a | = amplitude =
period = period =
b = b =
h = horizontal phase shift h = horizontal phase shift
k = vertical displacement

k = vertical displacement (midline)
k =

Unless otherwise stated, apply the expansions or compressions before applying the translations. Suggested application order:

  • a
    • vertical stretch about the x-axis
    • vertical reflection in the x-axis when a < 0
  • b
    • horizontal stretch about the y-axis
    • horizontal reflection in the y-axis when b < 0
  • h
    • horizontal translation
  • k
    • vertical translation

Interactive Activity

  • Click on the applet to activate
  • Click RESET. Click sin
  • Move the parameter sliders for a, b, h and k and study the effects.
  • Determine the parameter that effects:
    • vertical stretch (expansion/compression) /reflection
    • horizontal stretch (expansion/compression)/reflection
    • horizontal translation (shift left/right)
    • vertical translation (shift up/down)
  • Watch the equation change to see the position of the a, b, h and k parameters.

    y = a sin [ b( x - h) ] + k

  • Repeat for cos.

    y = a cos [ b( x - h) ] + k

y = a sin [ b( x - hpi) ] + k

or

(y - k) = sin [ b( x - hpi) ]
   a

y = a cos [ b( x - hppi) ] + k

or

(y - k) = cos [ b( x - hpi) ]
   a

Linked Source - Ron Blond

 

A copy of the above applet is provided to study the remainder of the notes for descriptions of stretches.

y = a sin [ b( x - hpi) ] + k

or

(y - k) = sin [ b( x - hpi) ]
   a

y = a cos [ b( x - hppi) ] + k

or

(y - k) = cos [ b( x - hpi) ]
   a

Linked

 

Vertical Stretch/Reflection

In both functions, a is the vertical stretch factor.

transformed function

y = a f (x) or y y/a

(x, y) → (x, ay)

vertical stretch factor (amplitude)

| a | = amplitude

      =

reflection

reflection in the x-axis when a < 0 and k = 0
reflection in y = k when a < 0 and k ≠ 0

Creates:

  • a f (x)
  • -f (x)
x-intercept(s)

no changes when y = 0 and k = 0.

Domain - set of possible x-values xR

Range - set of possible y-values

between crest/troughs: (k - a) < y < (k + a)

 

(h, k)

changing a does not change location of red data point, (h, k)

Description of Vertical Stretch

If a > 1, then a(x) is an expansion

  • When a = 3, then 3(x): vertical stretch by a factor of 3.

else if a = 1, then a(x) = x: No Expansion/Compression

else if 0 < a < 1, then a(x) is a compression

  • When a = 1/3, then 1/3(x) is a vertical stretch by a factor of 1/3.

else if a = 0, then a(x) = 0 is a linear graph of y = k

else if -1 < a < 0, then a(x) is a compression and reflection

  • When a = -1/3, then -1/3(x) is a vertical stretch by a factor of 1/3 and reflection in x-axis

else if a = -1, then -(x) is a reflection (in the x-axis) only

else if -a < -1, then a(x) is an expansion and reflection

  • When a = -3, then -3(x) is a vertical stretch by a factor of 3 and reflection in x-axis
Algebraic Description of Vertical Stretch

y y/a: Replace the y in the original function with y/a.

(x, y) (x, ay)

mapping notation to find location of data points after a vertical stretch.

Interactive Activity

  • Click on the applet to activate if needed.
  • Click RESET. Select sin
  • Move the parameter slider for a.
    • note the position of the x-intercept(s) and y-intercept
  • Repeat for cos.

y = a sin [ b( x - hpi) ] + k

or

(y - k) = sin [ b( x - hpi) ]
   a

y = a cos [ b( x - hppi) ] + k

or

(y - k) = cos [ b( x - hpi) ]
   a

Linked Source - Ron Blond

 

A copy of the above applet is provided to study the remainder of the notes for descriptions of stretches.

y = a sin [ b( x - hpi) ] + k

or

(y - k) = sin [ b( x - hpi) ]
   a

y = a cos [ b( x - hppi) ] + k

or

(y - k) = cos [ b( x - hpi) ]
   a

Linked Source - Ron Blond

Horizontal Stretch/Reflection

In both functions, 1/b is the horizontal stretch factor.

transformed function

y = f (bx) or x bx

(x, y) → (x/b, y)

If f [sin(4x-12)], rewrite as f [sin(4(x-3))]

b b =
horizontal stretch factor

1/b

period period =
reflection

reflection in the y-axis, when b < 0 and h = 0
reflection in y = k when, a < 0 and k ≠ 0

Creates:

  • f (bx)
  • f (-x).
y-intercept(s)

no changes when y = 0 and k = 0.

Domain - set of possible x-values xR

Range - set of possible y-values

( k- a) < y < (k + a) → y-values between crest/troughs

(h, k)

changing b does not change location of red data point, (h, k).

Description of Horizontal Stretch
  • If b > 1, then (bx) is a compression

when b = 3, then (3x) is a horizontal stretch by a factor of 1/3

  • else if b = 1, then (bx) = x and there is no expansion/compression
  • else if 0 < b < 1, then (bx) is an expansion

when b = 1/3, then (1/3 x) is a horizontal stretch by a factor of 3

  • else if b = 0, then (bx) = 0 is a linear graph of y = k
  • else if -1 < b < 0, then (bx) is an expansion and reflection

when b = -1/3, then (-1/3 x): horizontal stretch by a factor of 3 and reflection in the x-axis

  • else if b = -1, then (-x) is a reflection (in y-axis) only
  • else if b < -1, then a(x) is an expansion and reflection

when a = -3, then (-3x) is a horizontal stretch by a factor of 1/3 and reflection in x-axis

Algebraic Description of Horizontal Stretch

x bx: Replace the x in the original function with bx.

Note: Horizontal stretch factor is 1/b.

(x, y) (1/bx, y)

mapping notation to find location of data points after a horizontal stretch.

Interactive Activity

  • Click on the applet to activate if needed.
  • Click RESET. Select sin
  • Move the parameter slider for a.
    • note the position of the x-intercept(s) and y-intercept
  • Repeat for cos.

y = a sin [ b( x - hpi) ] + k

or

(y - k) = sin [ b( x - hpi) ]
   a

y = a cos [ b( x - hppi) ] + k

or

(y - k) = cos [ b( x - hpi) ]
   a

Linked Source - Ron Blond

 

A copy of the above applet is provided to study the remainder of the notes for descriptions of stretches.

y = a sin [ b( x - hpi) ] + k

or

(y - k) = sin [ b( x - hpi) ]
   a

y = a cos [ b( x - hppi) ] + k

or

(y - k) = cos [ b( x - hpi) ]
   a

 

Horizontal Translation (Slide)

In both functions, h is the horizontal translation parameter.

transformed function

y = f (x - h) or x x - h

(x, y) → (x + h, y)

horizontal translation

h

Domain - set of possible x-values xR

Range - set of possible y-values

(k - a) < y < (k + a) - between crest/troughs

(h, k)

Location of red data point on graph

Describing Horizontal Translation
  • If h > 0, then (x - h) is a translation h units right.

when h = 3, (x - 3) is a translation 3 units right

  • else if h = 0, then (x - 0) = (x) has no horizontal translation.
  • else if h < 0, then (x - h) is a translation h units left.

when h = -3, (x - -3) = (h + 3): translation 3 units left

Algebraic Description of Horizontal Translation/Shift

x arrow x - h

  • Replace the x in the original function with (x - h)

(x, y) (x + h, y)

mapping notation to find location of data points after a horizontal translation.

Interactive Activity

  • Click on the applet to activate if needed.
  • Click RESET. Select sin
  • Move the parameter slider for h.
  • Repeat for cos.

Linked Source - Ron Blond

 

Vertical Translation (Slide)

In both functions, k is the vertical translation parameter.

transformed function

y = f (x) + k or y y - k

(x, y) → (x, y + k)
vertical translation/ displacement (midline)

k =

Domain - set of possible x-values xR

Range - set of possible y-values

(k - a) < y < (k + a) - between crest/troughs

(h, k)

Location of red data point on graph

Description of Vertical Translation/Shift
  • If k > 0, then (x) + k is a translation k units up

when k = 3, then (x) + 3is a translation 3 units up

  • else if k = 0, then (x) - 0 = (x) has no vertical translation.
  • else if k < 0, then (x) + k is a translation k units down.

when k = -3, then (x) - 3: translation 3 units down

Algebraic Description of Vertical Translation/Shift

y y - k

  • Replace the y in the original function with (y - k)

(x, y) (x, y + k)

mapping notation to find location of data points after a vertical translation.

Interactive Activity

  • Click on the applet to activate if needed.
  • Click RESET. Select sin
  • Move the parameter slider for k.
  • Repeat for cos.

Linked Source - Ron Blond

Math 30P   Sine/Cosine   Translations    Stretches   Reflections   Inverses   Reciprocals   Combinations   Prerequisites

 
Comments to:  Jim Reed - Homepage
Started September, 1998. Copyright 2006, 2007