Transformations of Trigonometric Functions and Their Graphs

We are given a function that models the blood pressure P of a person at rest at time t. P=120-15 cos 5π/2 t The blood pressure P is given in millimeters of mercury and t is time in seconds. We want to graph this function. We will start by comparing it with the general form of a transformed cosine function. General Form y= a cos b(t- h)+ k [0.5em] Given Function P= -15 cos 5π/2 (t- 0)+ 120 As we can see, a= - 15, b= 5π2, h= 0, and k= 120. Now let's recall how these variables affect the graph of the parent cosine function.

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

Now we will transform the parent cosine function step by step. First, we will stretch the parent cosine function vertically by a factor of 15 and shrink it horizontally by a factor of 5π2, or 2.5 π. Let's calculate the amplitude and period of the cosine function. Amplitude: & | a| ⇒ | 15|=15 [0..5em] Period: & 2π/| b| ⇒ 2π/| 2.5 π| =0.8 Now we will draw the transformed function by using the amplitude and period that we found.

Great! Notice that a= - 15. Since it is less than 0, it also represents a reflection in the x-axis or across the midline y=0. Let's now reflect our graph in the x-axis!

We will next translate this function 120 units up. Since h= 0, there is no phase shift.

Finally, we will set the starting point of the graph at t=0 to find the desired graph because the time cannot be negative.

The graph matches option C.

We know that one cycle represents one heartbeat. To find the number of heartbeats per minute, we will examine the graph. From the graph we can tell that one cycle takes 0.8 seconds.

We want to determine the pulse rate of the person in heartbeats per minute. Because there are 60 seconds in 1 minute, we need to find the number of heartbeats in 60 seconds. Since one heartbeat takes 0.8 seconds, let's divide 60 by 0.8. 60/0.8=75 There are 75 heartbeats in 60 seconds. This means that the pulse rate is 75 heartbeats per minute.

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 $b$

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

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

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

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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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

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

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

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

Function	Reciprocal
y=sin x	y=csc x
y=cos x	y=sec x