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Transformations of Functions>
Combinations of Transformations: Order>
Combinations of Transformations: OrderCombinations of Transformations: Order
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Question 1 of 8
1. Question
When transforming y=x3 to y=3x3-2 is the vertical dilation factor of 3 or the vertical translation of 2 unit down applied first?
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The vertical dilation factor is applied first.
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Order is important when a vertical dilation is combined with a vertical translation.A standard function form for a vertical translation is y=f(x)+c where +c is an upward movement along the y-axis.A standard function form for a vertical dilation factor is y=kf(x) where k is the vertical dilation factor.In order to find the order of the transformations we will try. First, lets try a vertical dilation in order to transform y=x3 to y=3x3-2.The first transformation to be applied on y=x3 will be the vertical dilation factor k where k=3.y= 3x3 Apply the vertical dilation factor k=3. Remember y=kf(x). The second transformation to be applied on y=3x3 will be the vertical translation y=f(x)+c where c=–2.y= 3x3-2 Apply the vertical translation c=–2. Remember y=f(x)+c. The vertical dilation factor is applied first. -
Question 2 of 8
2. Question
When transforming y=x3 to y=(5x-1)3, which comes first. The horizontal dilation factor of 15 or the horizontal translation of 1 unit to the right?
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The horizontal dilation factor is applied first. -
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The horizontal translation is applied first.
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Order is important when a horizontal dilation is combined with a horizontal translation.A standard function form for a horizontal translation is y=f(x+h) where +h is a left movement along the x-axis.The application of the horizontal dilation factor on a x variable is factor=1a and we can apply xfactor.To find the correct order we will try one of two ways.To go from y=x3 to y=(5x-1)3, lets try the horizontal scale factor first.First let’s apply the Horizontal Scale Factor=15.y = x3 Then we use xfactor where we divide x by 15 y = (x15)3 Simplify. y = (5x)3 Secondly, we apply the horizontal translation of 1 unit to the rightWe will use y=f(x+h) where ++h is moving to the leftNotice h=-1 when we move to the rightWe replace x for (x-1) in y=(5x)3.It becomes y=(5(x-1))3This clearly is incorrect as it does not match y=(5x-1)3. Which means the order is incorrect.Now lets try another order.Starting with y=x3.First we will apply the horizontal translation h to y=x3 where h=-1 (moving to the right by 1 unit). Now we replace x for (x-1) in y=x3It simplifies to y=(x-1)3.The second transformation to be applied will be the horizontal scale factor where factor=15.We will apply xfactor which becomes y=(x15-1)3 this simplifies to:y=(5x-1)3Yes, this matches the original function!Correct order: first horizontal translation then horizontal dilationThe horizontal translation is applied first. -
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Question 3 of 8
3. Question
When transforming y=lnx to y=ln[4(x+5)] which comes first. The horizontal dilation factor of 14 or the horizontal translation of 5 units to the left?
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The horizontal dilation factor is applied first. -
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The horizontal translation is applied first.
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Order is important when a horizontal dilation is combined with a horizontal translation.A standard function form for a horizontal translation is y=f(x+h) where +h is a left movement along the x-axis.The application of the horizontal dilation factor on a x variable is factor=1a and we can apply xfactor.Lets try the order that the transformations will occur so y=lnx can become y=ln[4(x+5)].The first transformation to be applied to y=lnx will be the horizontal dilation factor=14.y= lnx We will apply xfactor, take the x and divide by 14. y= ln(x14) Simplify y= ln4x The second transformation to be applied to y=ln4x will be the horizontal translation h where h=+5 since it is moving to the left by 5 units.y= ln[4(x+5)] Apply the horizontal translation h=+5. Remember y=f(x+h). Yes, this is the correct order! It matches the original function.The horizontal dilation is applied first. -
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Question 4 of 8
4. Question
When transforming y=x2 to y=(15x+3)2 is the horizontal dilation factor of 5 or the horizontal translation of 3 units left applied first?
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The horizontal dilation factor is applied first. -
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The horizontal translation is applied first.
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Order is important when a horizontal dilation is combined with a horizontal translation.A standard function form for a horizontal translation is y=f(x+h) where +h is a left movement along the x-axis.The application of the horizontal dilation factor on a x variable is factor=1a and we can apply xfactor.To find the correct order where y=x2 becomes y=(15x+3)2 we are going to try a horizontal translation.The first transformation to be applied to y=x2 will be the horizontal translation h where h=+3.y= (x+3)2 Apply the horizontal translation h=+3. Remember y=f(x+h) where +h is moving to the left (opposite). The second transformation to be applied to y=(x+3)2 will be the horizontal dilation factor=5.y= (x5)+3)2 Apply the horizontal dilation factor=5. We apply xfactor by taking the x and dividing it by 5. y= (15x+3)2 Yes, this is the correct order! It matches.The horizontal translation is applied first. -
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Question 5 of 8
5. Question
Describe the order of the transformations when transforming y=ex to y=3e6x-1.
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1. Horizontal translation, 2. Vertical dilation, 3. Vertical translation -
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1. Vertical translation, 2. Horizontal dilation, 3. Vertical dilation -
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1. Horizontal dilation, 2. Vertical dilation, 3. Vertical translation -
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1. Horizontal dilation, 2. Vertical translation, 3. Vertical translation
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Order is important when a vertical dilation is combined with a vertical translation.A standard function form for a vertical translation is y=f(x)+c where c is an upward movement along the y-axis.The application of the horizontal dilation (factor) on a x variable is factor=1a and we can apply xfactor.A standard function form for a vertical dilation factor is y=kf(x) where k is the vertical dilation factor.To find the correct order so we can go from y=ex to y=3e6x-1, lets try a horizontal dilation.The first transformation to be applied to y=ex will be the horizontal dilation factor=1a where a=6.
Notice factor=16.This is from the given equation y=3e6x-1.This is in the form y=eax where a=6, so:y= e6x Alternatively, we could use xfactor which equals ex16 which equals: y=e6x.The second transformation to be applied on y=e6x will be the vertical dilation factor k where k=3.y= 3e6x Apply the vertical dilation factor k=3. Remember y=kf(x). The third transformation to be applied on y=3e6x will be the vertical translation y=f(x)+c where c=-1.y= 3e6x-1 Apply the vertical translation c=-1. Yes, this is the correct order! It matches the original function.1. Horizontal dilation, 2. Vertical dilation, 3. Vertical translation -
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Question 6 of 8
6. Question
Describe the order of the transformations when transforming y=logx to y=7log(x+5)-4.
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1. Vertical translation, 2. Vertical dilation, 3. Vertical translation -
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1. Horizontal translation, 2. Vertical dilation, 3. Vertical translation -
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1. Vertical translation, 2. Vertical dilation, 3. Horizontal translation -
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1. Horizontal translation, 2. Vertical dilation, 3. Vertical translation
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Order is important when a vertical dilation is combined with a vertical translation.A standard function form for a vertical translation is y=f(x)+c where +c is an upward movement along the y-axis.A standard function form for a horizontal translation is y=f(x+h) where h is a shift to the left along the x-axis.A standard function form for a vertical dilation factor is y=kf(x) where k is the vertical dilation factor.To find the correct order we will try.To go from y=logx to y=7log(x+5)-4 lets try a horizontal translation.The first transformation to be applied to y=logx will be the horizontal translation h where h=+5.y= log(x+5) Apply the horizontal translation h=+5. Remember y=f(x+h). The second transformation to be applied on y=log(x+5) will be the vertical dilation factor k where k=7.y= 7log(x+5) Apply the vertical dilation factor k=7. Remember y=kf(x). The third transformation to be applied on y=7log(x+5) will be the vertical translation y=f(x)+c where c=-4.y= 7log(x+5)-4 Apply the vertical translation c=-4. Remember y=f(x)+c. Yes, this is the correct order! It matches the original function.1. Horizontal translation, 2. Vertical dilation, 3. Vertical translation -
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Question 7 of 8
7. Question
Describe the order of the transformations when transforming y=|x| to y=|-4(x+2)|+6.
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1. Vertical translation 2. Horizontal dilation, 3. Horizontal translation, 4. Reflection about the y-axis -
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1. Reflection about the x-axis 2. Horizontal dilation, 3. Horizontal translation, 4. Vertical translation -
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1. Reflection about the y-axis 2. Vertical dilation, 3. Vertical translation, 4. Horizontal translation -
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1. Reflection about the y-axis 2. Horizontal dilation, 3. Horizontal translation, 4. Vertical translation
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Order is important when a horizontal dilation is combined with a horizontal translation.A standard function form for a vertical translation is y=f(x)+c where +c is an upward movement along the y-axis.A standard function form for a horizontal translation is y=f(x+h) where +h is a left movement along the x-axis.The application of the horizontal dilation (factor) on a x variable is factor=1a or we can use xfactor.A standard function form for a reflection about the y-axis is y=f(-x) where we replace x for -x.To find the correct order we will take a guess.To go from y=|x| to y=|-4(x+2)|+6 let’s try a reflection about the y-axis.The first transformation to be applied to y=|x| will be the reflection about the y-axis.y= |-x| Simply replace x for -x The second transformation to be applied to y=|-x| will be the horizontal dilation.We can use factor=1a.This is in the form y=|ax|. Where a=4Therefore y=|-4x|Alternatively, we can use factor=1a which equals 14.
Therefore horizontal factor=14Now using xfactor=|x14|Therefore y= |-4x| The third transformation to be applied to y=|-4x| will be the horizontal translation h where h=+2 (where it is 2 units to the left).y= |-4(x+2)| Apply the horizontal translation h=+2. Remember y=f(x+h). The fourth transformation to be applied on y=|-4(x+2)| will be the vertical translation y=f(x)+c where c=+6.y= |-4(x+2)|+6 Yes, this is the correct order! It matches the original function.1. Reflection about the y-axis 2. Horizontal dilation, 3. Horizontal translation, 4. Vertical translation -
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Question 8 of 8
8. Question
Describe the order of the transformations when transforming y=x3 to y=8(3x-9)3+2.
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1. Vertical dilation, 2. Vertical translation, 3. Horizontal dilation, 4. Horizontal translation -
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1. Horizontal translation, 2. Horizontal dilation, 3. Vertical dilation, 4 Vertical translation -
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1. Vertical dilation, 2. Horizontal translation, 3. Horizontal dilation, 4 Vertical translation -
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1. Horizontal dilation, 2. Horizontal translation, 3. Vertical dilation, 4. Vertical translation
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Order is important is when horizontal dilation is combined with a horizontal translation. Order is also important when a vertical dilation is combined with a vertical translationA standard function form for a vertical translation is y=f(x)+c where +c is an upward movement along the y-axis.A standard function form for a horizontal translation is y=f(x+h) where +h is a shift to the left along the x-axis.A standard function form for a vertical dilation factor is y=kf(x) where k is the vertical dilation factor.The application of the horizontal dilation factor on a x variable is factor=1a or we can apply xfactor.The universal formula form with two translations and two dilations factors in one equation is y=kf(a(x+b))+c, where b is also h in the horizontal translation.
k is the vertical dilation factor. +b is shift left. +c is shift up. Horizontal factor is 1a.To find the correct order we will take a guess to go from y=x3 to y=8(3x-9)3+2. First we are going to factor it out.y = 8(3(x-3))3+2 Factor out 3 from 3x-9. Now it is in the universal formula form y=kf(a(x+b))+c. The first transformation to be applied to y=x3 will be the horizontal dilation factor=1a where a=3 (from universal formula).y= (3x)3 Apply the horizontal parameter a=3. Remember factor=1a and xfactor. The second transformation to be applied to y=(3x)3 will be the horizontal translation h where h=-3. Simply replace x for x-3.y= (3(x–3))3 Apply the horizontal translation h=-3. Remember y=f(x+h). The third transformation to be applied on y=(3(x-3))3 will be the vertical dilation factor k where k=8.y= 8(3(x-3))3 Apply the vertical dilation factor k=8. Remember y=kf(x). The fourth transformation to be applied on y=8(3(x-3))3 will be the vertical translation y=f(x)+c where c=+2.y= 8(3(x-3))3+2 Apply the vertical translation c=2. Remember y=f(x)+c. Yes, this is the correct order! It matches the original function.1. Horizontal dilation, 2. Horizontal translation, 3. Vertical dilation, 4. Vertical translation -
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Quizzes
- Vertical Translations 1
- Vertical Translations 2
- Vertical Translations from a Point
- Horizontal Translations 1
- Horizontal Translations 2
- Horizontal Translations from a Point
- Horizontal Translations from a Graph
- Horizontal and Vertical Translations from a Graph
- Sketch a Graph using Translations
- Write the Equation from a Graph
- Write the Equation from Translations
- Vertical Dilations
- Horizontal Dilations 1
- Horizontal Dilations 2
- Horizontal Dilations – Scale Factor
- Horizontal and Vertical Dilations 1
- Horizontal and Vertical Dilations 2
- Horizontal and Vertical Dilations 3
- Graphing Reflections 1
- Graphing Reflections 2
- Reflection with Rotation
- Combinations of Transformations: Order
- Combinations of Transformations: Coordinates
- Combinations of Transformations: Find Equation 1
- Combinations of Transformations: Find Equation 2
- Combinations of Transformations: Find Equation 3