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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 | ||
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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) | |
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Vertical Stretch or Shrink | If a>1,h=sint stretches vertically by ah=asint
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If 0<a<1,h=sint shrinks vertically by ah=asint
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Horizontal Stretch or Shrink | If b>1,h=sint shrinks horizontally by bh=sin(bt)
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If 0<b<1,h=sint stretches horizontally by bh=sin(bt)
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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) | |
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Period | ∣b∣π |
Asymptotes | 2∣b∣π |
Transformations of y=tanθ to y=atan(bθ) | |
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Vertical Stretch or Shrink | If a>1,y=tanθ stretches vertically by ay=atanθ
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If 0<a<1,y=tanθ shrinks vertically by ay=atanθ
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Horizontal Stretch or Shrink | If b>1,y=tanθ shrinks horizontally by by=tan(bθ)
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If 0<b<1,h=sinθ stretches horizontally by bh=sin(bθ)
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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 | |
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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 | |
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Vertical Translation | If k>0,y=tanθ moves k units upy=tan(x)+k
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If k<0,y=tanθ moves k units downy=tan(x)−k
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Horizontal Translation | If h>0,y=tanθ moves h units to the righty=tan(x−h)
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If h<0,y=tanθ moves h units to the lefty=tan(x+h)
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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 | |
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Vertical Stretch or Shrink | y=asinxIf a>1,y=sinx stretches vertically by a
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y=asinxIf 0<a<1,y=sinx shrinks vertically by a
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Horizontal Stretch or Shrink | y=sin(bx)If b>1y=sinx shrinks horizontally by b
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y=sin(bx)If 0<b<1y=sinx stretches horizontally by b
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Vertical Translation | y=sin(x)+kIf k>0,y=sinx moves k units up
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y=sin(x)+kIf k<0,y=sinx moves k units downy=sin(x)−k
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Horizontal Translation | y=sin(x−h)If h>0,y=sinx moves h units to the right
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y=sin(x−h)If h<0,y=sinx moves h units to the left
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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.