 {{ 'mllessonnumberslides'  message : article.intro.bblockCount}} 
 {{ 'mllessonnumberexercises'  message : article.intro.exerciseCount}} 
 {{ 'mllessontimeestimation'  message }} 
Here are a few recommended readings before getting started with this lesson.
Topics Related to Transformations of Functions
Topics Related to Trigonometric Functions
Like other functions, the parent functions of sine, cosine, and tangent can be transformed. The graphs of $y=asin(bx),$ $y=acos(bx),$ and $y=atan(bx)$ represent stretch or shrink transformations of their parent functions.
The graph of a parent trigonometric function can be vertically stretched or shrunk by multiplying the function rule by a constant, positive number $a.$ If $a>1,$ the function will be stretched. Conversely, the function will be shrunk if $0<a<1.$
Vertical Stretch or Shrink  

Parent Function  $a>1$  $0<a<1$ 
$y=sinx$  $y=asinx$ Stretch parent function vertically by a factor of $a$ 
$y=asinx$ Shrink parent function vertically by a factor of $a$ 
$y=cosx$  $y=acosx$ Stretch parent function vertically by a factor of $a$ 
$y=acosx$ Shrink parent function vertically by a factor of $a$ 
$y=tanx$  $y=atanx$ Stretch parent function vertically by a factor of $a$ 
$y=atanx$ Shrink parent function vertically by a factor of $a$ 
The graph of a trigonometric function can be horizontally stretched or shrunk by multiplying the input of the function by a positive number $b.$ If $b>1,$ the graph shrinks horizontally by a factor of $b.$ Conversely, if $0<b<1,$ the graph stretch horizontally by a factor of $b.$
Horizontal Stretch or Shrink  

Parent Function  $0<b<1$  $b>1$ 
$y=sinx$  $y=sin(bx)$ Stretch parent function horizontally by a factor of $b$ 
$y=sin(bx)$ Shrink parent function horizontally by a factor of $b$ 
$y=cosx$  $y=cos(bx)$ Stretch parent function horizontally by a factor of $b$ 
$y=cos(bx)$ Shrink parent function horizontally by a factor of $b$ 
$y=tanx$  $y=tan(bx)$ Stretch parent function horizontally by a factor of $b$ 
$y=tan(bx)$ Shrink parent function horizontally by a factor of $b$ 
Graph:
$b=0.16$
$∣0.16∣=0.16$
Use a calculator
Round to $1$ decimal place(s)
Transformations of $h=sint$ to $h=asin(bt)$  

Vertical Stretch or Shrink  $Ifa>1,h=sintstretches vertically byah=asint $

$If0<a<1,h=sintshrinks vertically byah=asint $
 
Horizontal Stretch or Shrink  $Ifb>1,h=sintshrinks horizontally bybh=sin(bt) $

$If0<b<1,h=sintstretches horizontally bybh=sin(bt) $

After going on the Ferris wheel, Dominika and Paulina go up to the big red triangular tower to see the townscape from the viewing deck.
Paulina wonders how to find the height of the tower. Dominika says that the height $y,$ in meters, of a tower like this one can be modeled by the following tangent function, where $θ$ is the angle indicated.Period: $32π $
$y=atan(bθ)$  

Period  $∣b∣π $ 
Asymptotes  $2∣b∣π $ 
Transformations of $y=tanθ$ to $y=atan(bθ)$  

Vertical Stretch or Shrink  $Ifa>1,y=tanθstretches vertically byay=atanθ $

$If0<a<1,y=tanθshrinks vertically byay=atanθ $
 
Horizontal Stretch or Shrink  $Ifb>1,y=tanθshrinks horizontally byby=tan(bθ) $

$If0<b<1,h=sinθstretches horizontally bybh=sin(bθ) $

$θ=50_{∘}$
Multiply
Use a calculator
Round to $2$ decimal place(s)
Trigonometric functions can be translated vertically or horizontally like the other functions. Next, these translations will be examined one at a time.
Let $f(x)$ be a parent trigonometric function. Then, $f(x)+k$ will translate the parent function vertically. If $k>0,$ then the graph moves $k$ units up. However, if $k<0,$ then the graph moves $k$ units down.
Vertical Translations  

Parent Function  $k>0$  $k<0$ 
$y=sinx$  $y=sinx+k,$ Translation $k$ units up 
$y=sinx−k,$ Translation $k$ units down 
$y=cosx$  $y=cosx+k,$ Translation $k$ units up 
$y=cosx−k,$ Translation $k$ units down 
$y=tanx$  $y=tanx+k,$ Translation $k$ units up 
$y=tanx−k,$ Translation $k$ units down 
A horizontal translation of a periodic function is called a phase shift. The graph of $f(x−h)$ represents a horizontal translation of $f(x)$ by $h$ units. For example, consider the parent functions of sine, cosine, and tangent functions. If $h>0,$ the parent trigonometric function will be shifted to the right $h$ units, while if $h<0,$ the function will be shifted to the right $h$ units.
Horizontal Translations  

Parent Function  $h>0$  $h<0$ 
$y=sinx$  $y=sin(x−h)$ Translation $h$ units to the right 
$y=sin(x+h)$ Translation $h$ units to the left 
$y=cosx$  $y=cos(x−h)$ Translation $h$ units to the right 
$y=cos(x+h)$ Translation $h$ units to the left 
$y=tanx$  $y=tan(x−h)$ Translation $h$ units to the right 
$y=tan(x+h)$ Translation $h$ units to the left 
Now Dominika and Paulina are waiting to ride the carousel in the amusement park. The girls enjoy the up and down movement of wooden horses.
Paulina thinks that the movement of a rider on a horse looks like a cosine function. While talking to the operator of the ride, the girls learn that each horse has a special route. They choose horses whose routes can be represented by the following functions.Write in the Form $y=acosb(x−h)+k$  Amplitude: $∣a∣$  Period: $∣b∣2π $  

$y=cosx$  $y=1cos1(x−0)+0$  $∣1∣=1$  $∣1∣2π =2π$ 
$y=cos(x+π)−1$  $y=1cos1(x−(π))+(1)$  $∣1∣=1$  $∣1∣2π =2π$ 
After riding on the carousel, Dominika and Paulina are looking for something more exciting when they hear the screams coming from the roller coaster. They watch the roller coaster for a while, then see the section where people are most excited and scream the loudest.
The track in this section is quite steep. The girls do some quick math and determine that the path of the roller coaster can be modeled by the following tangent function.Translations of $y=tanθ$ to $y=tan(θ−h)+k$  

Vertical Translation  $Ifk>0,y=tanθmoves k units upy=tan(x)+k $

$Ifk<0,y=tanθmoves k units downy=tan(x)−k $
 
Horizontal Translation  $Ifh>0,y=tanθmoves h units to the righty=tan(x−h) $

$Ifh<0,y=tanθmoves h units to the lefty=tan(x+h) $

From the table, it can be concluded that the function $y=tan(x−3π)+1$ results from shifting the parent tangent function $3π$ units to the right and $1$ unit up.
After the roller coaster, Dominika and Paulina decided to take a rest. They see a math game where each winner is awarded a teddy bear and think that it may be fun to take a look at the question.
Dominika and Paulina join the game individually and each give different answers to the question.Identify the parent function and apply one transformation at a time.
Transformations of $y=sinx$  

Vertical Stretch or Shrink  $y=asinxIfa>1,y=sinxstretches vertically bya $

$y=asinxIf0<a<1,y=sinxshrinks vertically bya $
 
Horizontal Stretch or Shrink  $y=sin(bx)Ifb>1y=sinxshrinks horizontally byb $

$y=sin(bx)If0<b<1y=sinxstretches horizontally byb $
 
Vertical Translation  $y=sin(x)+kIfk>0,y=sinxmoves k units up $

$y=sin(x)+kIfk<0,y=sinxmoves k units downy=sin(x)−k $
 
Horizontal Translation  $y=sin(x−h)Ifh>0,y=sinxmoves h units to the right $

$y=sin(x−h)Ifh<0,y=sinxmoves h units to the left $

Substitute values
$a−(b)=a+b$
Dominika and Paulina have had fun during the day in the amusement park. They decide to come again together next week. To choose an appropriate day, they check the weather for the next two weeks.