Sine and Cosine Functions

Discover and remember the pattern(s) by completing Help to Remember What to Learn

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

What to Learn

Help to Remember What to Learn

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 - 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

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

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 - 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

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 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 - 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

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) 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 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.

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.

Comments to:  Jim Reed - Homepage