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

Sine and Cosine Functions

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Standard Form

Parameters:

a

(amplitude) vertical stretch/reflection if a < 0

  • changes range, but not domain
1/b

horizontal stretch/reflection if b < 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 about the x-axis
  • b
    • horizontal stretch about the y-axis
    • horizontal reflection about the y-axis
  • 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)
  • Repeat for cos.

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

or

(y - k) = sin [ b( x - h) ]
    a

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

or

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

or

(y - k) = sin [ b( x - h) ]
    a

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

or

(y - k) = cos [ b( x - h) ]
    a

Linked Source - Ron Blond

 

Vertical Stretch/Reflection

In both functions, a is the vertical stretch factor.

transformed function

y = af(x) or y y/a

(x, y) → (x, ay)
vertical stretch factor (amplitude)

| a | = amplitude

      =

reflection

about x-axis when a < 0 and k = 0. Creates:

  • af(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

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

 

(h, k)

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

If a > 1, then a(x): 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): Compression

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

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

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

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

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

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

  • When a = -3, then -3(x): 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 - h) ] + k

or

(y - k) = sin [ b( x - h) ]
    a

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

or

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

or

(y - k) = sin [ b( x - h) ]
    a

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

or

(y - k) = cos [ b( x - h) ]
    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

about x-axis when b < 0 and h = 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): Compression

  • When b = 3, then (3x): horizontal stretch by a factor of 1/3.

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

else if 0 < b < 1, then (bx): Expansion

  • When b = 1/3, then (1/3 x): horizontal stretch by a factor of 3.

else if b = 0, then (bx) = 0: linear graph of y = k

else if-1 < b < 0, then (bx): expansion/reflection

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

else if b = -1, then (-x): reflection (in x-axis) only

else if-b < -1, then a(x): expansion/reflection

  • When a = -3, then (-3x): 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 - h) ] + k

or

(y - k) = sin [ b( x - h) ]
    a

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

or

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

or

(y - k) = sin [ b( x - h) ]
    a

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

or

(y - k) = cos [ b( x - h) ]
    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): translation h units right.

  • When h = 3, then (x - 3): translation 3 units right

else if h = 0, then (x - 0) = (x): no horizontal translation.

else if h < 0, then (x - h): translation h units left.

  • When h = -3, then (x - -3) = (h + 3): translation 3 units left
Algebraic Description of Horizontal Translation/Shift

x 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: translation k units up

  • When k = 3, then (x) + 3: translation 3 units up

else if k = 0, then (x) - 0 = (x): no vertical translation.

else if k < 0, then (x) + k: 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