Understanding Transformations, Amplitude, and Period in Trigonometric Functions

Vertical Stretch or Shrink
Parent Function	$a > 1$	$0 < a < 1$
$y = sin x$	$y = a sin x$ Stretch parent function vertically by a factor of $a$	$y = a sin x$ Shrink parent function vertically by a factor of $a$
$y = cos x$	$y = a cos x$ Stretch parent function vertically by a factor of $a$	$y = a cos x$ Shrink parent function vertically by a factor of $a$
$y = tan x$	$y = a tan x$ Stretch parent function vertically by a factor of $a$	$y = a tan x$ Shrink parent function vertically by a factor of $a$

Vertical Stretch or Shrink

Parent Function

a > 1

0 < a < 1

y = sin x

y = a sin x

Stretch parent function vertically
by a factor of

a

y = a sin x

Shrink parent function vertically
by a factor of

a

y = cos x

y = a cos x

Stretch parent function vertically
by a factor of

a

y = a cos x

Shrink parent function vertically
by a factor of

a

y = tan x

y = a tan x

Stretch parent function vertically
by a factor of

a

y = a tan x

Shrink parent function vertically
by a factor of

a

Horizontal Stretch or Shrink
Parent Function	$0 < b < 1$	$b > 1$
$y = sin x$	$y = sin (b x)$ Stretch parent function horizontally by a factor of $b$	$y = sin (b x)$ Shrink parent function horizontally by a factor of $b$
$y = cos x$	$y = cos (b x)$ Stretch parent function horizontally by a factor of $b$	$y = cos (b x)$ Shrink parent function horizontally by a factor of $b$
$y = tan x$	$y = tan (b x)$ Stretch parent function horizontally by a factor of $b$	$y = tan (b x)$ Shrink parent function horizontally by a factor of $b$

Horizontal Stretch or Shrink

Parent Function

0 < b < 1

b > 1

y = sin x

y = sin (b x)

Stretch parent function horizontally
by a factor of

b

y = sin (b x)

Shrink parent function horizontally
by a factor of

b

y = cos x

y = cos (b x)

Stretch parent function horizontally
by a factor of

b

y = cos (b x)

Shrink parent function horizontally
by a factor of

b

y = tan x

y = tan (b x)

Stretch parent function horizontally
by a factor of

b

y = tan (b x)

Shrink parent function horizontally
by a factor of

b

Transformations of $h = sin t$ to $h = a sin (b t)$
Vertical Stretch or Shrink	$If a > 1, h = sin t stretches vertically by a h = a sin t$
$If 0 < a < 1, h = sin t shrinks vertically by a h = a sin t$
Horizontal Stretch or Shrink	$If b > 1, h = sin t shrinks horizontally by b h = sin (b t)$
$If 0 < b < 1, h = sin t stretches horizontally by b h = sin (b t)$

Transformations of

h = sin t

h = a sin (b t)

Vertical Stretch or Shrink

If a > 1, h = sin t stretches vertically by a h = a sin t

If 0 < a < 1, h = sin t shrinks vertically by a h = a sin t

Horizontal Stretch or Shrink

If b > 1, h = sin t shrinks horizontally by b h = sin (b t)

If 0 < b < 1, h = sin t stretches horizontally by b h = sin (b t)

	$y = a tan (b θ)$
Period	$\frac{π}{∣ b ∣}$
Asymptotes	$\frac{π}{2 ∣ b ∣}$

y = a tan (b θ)

Period

\frac{π}{∣ b ∣}

Asymptotes

\frac{π}{2 ∣ b ∣}

Transformations of $y = tan θ$ to $y = a tan (b θ)$
Vertical Stretch or Shrink	$If a > 1, y = tan θ stretches vertically by a y = a tan θ$
$If 0 < a < 1, y = tan θ shrinks vertically by a y = a tan θ$
Horizontal Stretch or Shrink	$If b > 1, y = tan θ shrinks horizontally by b y = tan (b θ)$
$If 0 < b < 1, h = sin θ stretches horizontally by b h = sin (b θ)$

Transformations of

y = tan θ

y = a tan (b θ)

Vertical Stretch or Shrink

If a > 1, y = tan θ stretches vertically by a y = a tan θ

If 0 < a < 1, y = tan θ shrinks vertically by a y = a tan θ

Horizontal Stretch or Shrink

If b > 1, y = tan θ shrinks horizontally by b y = tan (b θ)

If 0 < b < 1, h = sin θ stretches horizontally by b h = sin (b θ)

Vertical Translations
Parent Function	$k > 0$	$k < 0$
$y = sin x$	$y = sin x + k,$ Translation $k$ units up	$y = sin x - k,$ Translation $k$ units down
$y = cos x$	$y = cos x + k,$ Translation $k$ units up	$y = cos x - k,$ Translation $k$ units down
$y = tan x$	$y = tan x + k,$ Translation $k$ units up	$y = tan x - k,$ Translation $k$ units down

Vertical Translations

Parent Function

k > 0

k < 0

y = sin x

y = sin x + k,

Translation

k

units up

y = sin x - k,

Translation

k

units down

y = cos x

y = cos x + k,

Translation

k

units up

y = cos x - k,

Translation

k

units down

y = tan x

y = tan x + k,

Translation

k

units up

y = tan x - k,

Translation

k

units down

Horizontal Translations
Parent Function	$h > 0$	$h < 0$
$y = sin x$	$y = sin (x - h)$ Translation $h$ units to the right	$y = sin (x + h)$ Translation $h$ units to the left
$y = cos x$	$y = cos (x - h)$ Translation $h$ units to the right	$y = cos (x + h)$ Translation $h$ units to the left
$y = tan x$	$y = tan (x - h)$ Translation $h$ units to the right	$y = tan (x + h)$ Translation $h$ units to the left

Horizontal Translations

Parent Function

h > 0

h < 0

y = sin x

y = sin (x - h)

Translation

h

units to the right

y = sin (x + h)

Translation

h

units to the left

y = cos x

y = cos (x - h)

Translation

h

units to the right

y = cos (x + h)

Translation

h

units to the left

y = tan x

y = tan (x - h)

Translation

h

units to the right

y = tan (x + h)

Translation

h

units to the left

	Write in the Form $y = a cos b (x - h) + k$	Amplitude: $∣ a ∣$	Period: $\frac{2 π}{∣ b ∣}$
$y = cos x$	$y = 1 cos 1 (x - 0) + 0$	$∣ 1 ∣ = 1$	$\frac{2 π}{∣ 1 ∣} = 2 π$
$y = cos (x + π) - 1$	$y = 1 cos 1 (x - (- π)) + (- 1)$	$∣ 1 ∣ = 1$	$\frac{2 π}{∣ 1 ∣} = 2 π$

Write in the Form

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

Amplitude:

∣ a ∣

Period:

\frac{2 π}{∣ b ∣}

y = cos x

y = 1 cos 1 (x - 0) + 0

∣ 1 ∣ = 1

\frac{2 π}{∣ 1 ∣} = 2 π

y = cos (x + π) - 1

y = 1 cos 1 (x - (- π)) + (- 1)

∣ 1 ∣ = 1

\frac{2 π}{∣ 1 ∣} = 2 π

Translations of $y = tan θ$ to $y = tan (θ - h) + k$
Vertical Translation	$If k > 0, y = tan θ moves k units up y = tan (x) + k$
$If k < 0, y = tan θ moves k units down y = tan (x) - k$
Horizontal Translation	$If h > 0, y = tan θ moves h units to the right y = tan (x - h)$
$If h < 0, y = tan θ moves h units to the left y = tan (x + h)$

Translations of

y = tan θ

y = tan (θ - h) + k

Vertical Translation

If k > 0, y = tan θ moves k units up y = tan (x) + k

If k < 0, y = tan θ moves k units down y = tan (x) - k

Horizontal Translation

If h > 0, y = tan θ moves h units to the right y = tan (x - h)

If h < 0, y = tan θ moves h units to the left y = tan (x + h)

Transformations of

y = sin x

Vertical Stretch or Shrink

y = a sin x If a > 1, y = sin x stretches vertically by a

y = a sin x If 0 < a < 1, y = sin x shrinks vertically by a

Horizontal Stretch or Shrink

y = sin (b x) If b > 1 y = sin x shrinks horizontally by b

y = sin (b x) If 0 < b < 1 y = sin x stretches horizontally by b

Vertical Translation

y = sin (x) + k If k > 0, y = sin x moves k units up

y = sin (x) + k If k < 0, y = sin x moves k units down y = sin (x) - k

Horizontal Translation

y = sin (x - h) If h > 0, y = sin x moves h units to the right

y = sin (x - h) If h < 0, y = sin x moves h units to the left

a To write the sine function, start by drawing the function. To do so, plot the days and the corresponding temperatures as points in the coordinate plane.

Since the curve connecting the points looks like a sinusoid, its period, midline, and amplitude can be used to write a sine function. Note that the graph moves periodically along the

x -

axis and one full cycle is completed from

x = 0

x = 14 .

This means that the period of the function can be assumed to be

14 .

Period : 14

The maximum and minimum values of the function are

95

and

45,

respectively. The amplitude can be determined by finding half of the difference of these values.

Amplitude : \frac{9 5 - 4 5}{2} = 25

The amplitude is

25 .

Finally, the midline will be calculated as a mean of the minimum and maximum values.

Midline : \frac{9 5 + 4 5}{2} = 70

The sine function can now be written by using all of this information. Start by recalling the general transformed form of a sine function.

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

In this form,

∣ a ∣

is the amplitude. According to graph, the amplitude is

25 .

∣ a ∣ = 25 ⇓ a = 25 or a = - 25

To determine the sign of

a,

compare its graph with the graph of parent sine function.

Notice that after intersecting the midline at

x = 0,

the graph goes down to its minimum and then goes up to its maximum, unlike the parent sine function, which goes up at first and then down. Therefore, this graph represents a reflection in the midline. Therefore, the value of

a

needs to be negative.

a = - 25

Recall that the formula for the period of a sine function is

\frac{2 π}{∣ b ∣} .

The value of

b

in the formula rule can be calculated by setting the period of the graph,

p = 14 .

p = \frac{2 π}{∣ b ∣}

Substitute

$p = 14$

14 = \frac{2 π}{∣ b ∣}

▼

Solve for

b

MultEqn

$LHS \cdot ∣ b ∣ = RHS \cdot ∣ b ∣$

14 ∣ b ∣ = 2 π

DivEqn

$LHS / 14 = RHS / 14$

∣ b ∣ = \frac{2 π}{1 4}

AbsPos

$∣ b ∣ = b$

b = \frac{2 π}{1 4}

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

b = \frac{π}{7}

Since one full cycle of the graph starts from

x = 0

and ends at

x = 14,

which is the interval of exactly one period, it can be said that there is no phase shift. Therefore,

h = 0 .

Finally, since the midline is

y = 70,

this refers that

k = 70 .

Now that all of the values for the variables are known, the sine function can be written.

y = a sin b (x - h) + k ⇓ y = - 25 sin \frac{π}{7} (x - 0) + 70 ⇕ y = - 25 sin \frac{π}{7} x + 70

The given forecast over the next two weeks can be modeled by the sine function

y = - 25 sin \frac{π}{7} x + 70 .

b To determine the types of transformations applied to the parent sine function, start by recalling what the corresponding variables in the general form of the transformed sine function represent.

Transformations of $y = sin x$
Vertical Stretch or Shrink	$y = a sin x If a > 1, y = sin x stretches vertically by a$
Vertical Stretch or Shrink	$y = a sin x If 0 < a < 1, y = sin x shrinks vertically by a$
Horizontal Stretch or Shrink	$y = sin (b x) If b > 1, y = sin x shrinks horizontally by b$
Horizontal Stretch or Shrink	$y = sin (b x) If 0 < b < 1, y = sin x stretches horizontally by b$
Vertical Translation	$y = sin (x) + k If k > 0, y = sin x moves k units up$
Vertical Translation	$y = sin (x) - k If k < 0, y = sin x moves k units down$
Horizontal Translation	$y = sin (x - h) If h > 0, y = sin x moves h units to the right$
Horizontal Translation	$y = sin (x + h) If h < 0, y = sin x moves h units to the left$
Reflection	$y = - a sin x If a < 0, y = sin x reflects in the midline y = k$

Now recall the function rule written in Part A again.

y = - 25 sin \frac{π}{7} x + 70

According to the table, it can be said that the function rule represents a vertical stretch by a factor of

25,

a horizontal stretch by a factor of

\frac{π}{7},

and a vertical translation of parent sine function

70

units up. Also, since

a = - 25

is less than

0,

there is a reflection in the midline

y = 70 .

Parent Function	Transformed Form of the Function
$y = cot x$	$y = a cot b (x - h) + k$
$y = sec x$	$y = a sec b (x - h) + k$
$y = csc x$	$y = a csc b (x - h) + k$

Parent Function

Transformed Form of the Function

y = cot x

y = a cot b (x - h) + k

y = sec x

y = a sec b (x - h) + k

y = csc x

y = a csc b (x - h) + k

Transformations of Trigonometric Functions and Their Graphs

Catch-Up and Review

Stretching and Shrinking a Graph

Stretching and Shrinking Trigonometric Functions

Vertical Shrink or Stretch

Horizontal Shrink or Stretch

Ferris Wheel

Answer

Hint

Solution

Townscape

Answer

Hint

Solution

Translating a Trigonometric Function Graph

Translations of Trigonometric Functions

Vertical Translation

Horizontal Translation, or Phase Shift

Carousel

Hint

Solution

Roller Coaster

Hint

Solution

Math Game

Hint

Solution

Reflections of Parent Trigonometric Functions

Reflecting Trigonometric Functions

Let's Meet Again

Answer

Hint

Solution

Transformations of Cotangent, Secant, and Cosecant Functions

Transformations of Trigonometric Functions and Their Graphs

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