Combinations of Transformations: Order

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Question 1 of 8

1. Question
When transforming $y=x^3$ to $y=3x^3 -2$ is the vertical dilation factor of $3$ or the vertical translation of $2$ unit down applied first?
- 1. The vertical translation is applied first.
- 2. 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 $y=f(x)+color(purple)(c)$ where $color(purple)(+c)$ is an upward movement along the $y$ -axis.

A standard function form for a vertical dilation factor is $y=color(blue)(k)f(x)$ where $color(blue)(k)$ 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 $y = x^3$ to $y= 3x^3 -2$ .

The first transformation to be applied on $y=x^3$ will be the vertical dilation factor $color(blue)(k)$ where $color(blue)(k=3)$ .

$y=$ $color(blue)(3)x^3$ Apply the vertical dilation factor $color(blue)(k=3)$ . Remember $y=color(blue)(k)f(x)$ .

The second transformation to be applied on $y=3x^3$ will be the vertical translation $y=f(x)+color(purple)(c)$ where $color(purple)(c= – 2)$ .

$y=$ $3x^3 color(purple)(- 2)$ Apply the vertical translation $color(purple)(c= – 2)$ . Remember $y=f(x)+color(purple)(c)$ .

The vertical dilation factor is applied first.
Question 2 of 8

2. Question
When transforming $y=x^3$ to $y=(5x-1)^3$ , which comes first. The horizontal dilation factor of $1/5$ or the horizontal translation of $1$ unit to the right?
- 1. The horizontal translation is applied first.
- 2. The horizontal dilation factor 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 $y=f(x+color(red)(h))$ where $color(red)(+h)$ is a left movement along the $x$ -axis.

The application of the horizontal dilation $color(blue)(text{factor})$ on a $x$ variable is $color(blue)(text{factor})=1/color(green)(a)$ and we can apply $x/color(blue)(text{factor})$ .

To find the correct order we will try one of two ways.

To go from $y=x^3$ to $y=(5x-1)^3$ , lets try the horizontal scale factor first.

First let’s apply the Horizontal Scale $color(blue)(\text(Factor)=1/5)$ .

$y$ $=$ $x^3$ Then we use $x/\text(factor)$ where we divide $x$ by $color(blue)(1/5)$

$y$ $=$ $(x/(color(blue)(1/5)))^3$ Simplify.

$y$ $=$ $(5x)^3$

Secondly, we apply the horizontal translation of $1$ unit to the right

We will use $y=f(x color(red)(+h))$ where $+color(red)(+h)$ is moving to the left

Notice $color(red)(h=-1)$ when we move to the right

We replace $x$ for $(x color(red)(-1))$ in $y=(5x)^3$ .

It becomes $y=(5(x color(red)(-1)))^3$

This 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 $y=x^3$ .

First we will apply the horizontal translation $h$ to $y=x^3$ where $color(red)(h=-1)$ (moving to the right by $1$ unit). Now we replace $x$ for $(x color(red)(-1))$ in $y=x^3$

It simplifies to $y=(x color(red)(-1))^3$ .

The second transformation to be applied will be the horizontal scale factor where $color(blue)(\text(factor)=1/5)$ .

We will apply $x/\text(factor)$ which becomes $y=(x/(color(blue)(1/5))color(red)(-1))^3$ this simplifies to:

$y=(5x color(red)(-1))^3$

Yes, this matches the original function!

Correct order: first horizontal translation then horizontal dilation

The horizontal translation is applied first.
Question 3 of 8

3. Question
When transforming $y=lnx$ to $y=ln[4(x+5)]$ which comes first. The horizontal dilation factor of $1/4$ or the horizontal translation of $5$ units to the left?
- 1. The horizontal translation is applied first.
- 2. The horizontal dilation factor 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 $y=f(x+color(red)(h))$ where $color(red)(+h)$ is a left movement along the $x$ -axis.

The application of the horizontal dilation $color(blue)(text{factor})$ on a $x$ variable is $color(blue)(text{factor})=1/color(green)(a)$ and we can apply $x/color(blue)(text{factor})$ .

Lets try the order that the transformations will occur so $y=ln x$ can become $y=ln[4(x+5)]$ .

The first transformation to be applied to $y=lnx$ will be the horizontal dilation $color(blue)(text{factor}=1/4)$ .

$y=$ $ln x$ We will apply $x/color(blue)(text{factor})$ , take the $x$ and divide by $1/4$ .

$y=$ $ln (x/(1/4))$ Simplify

$y=$ $ln 4x$

The second transformation to be applied to $y=ln 4x$ will be the horizontal translation $color(red)(h)$ where $color(red)(h=+5)$ since it is moving to the left by $5$ units.

$y=$ $ln[4(x + color(red)(5))]$ Apply the horizontal translation $color(red)(h=+5)$ . Remember $y=f(x+color(red)(h))$ .

Yes, this is the correct order! It matches the original function.

The horizontal dilation is applied first.

Question 4 of 8

When transforming $y=x^2$ to $y=(1/5x + 3)^2$ is the horizontal dilation factor of $5$ or the horizontal translation of $3$ units left applied first?

1. The horizontal translation is applied first.
2. The horizontal dilation factor is applied first.

Question 5 of 8

5. Question
Describe the order of the transformations when transforming $y=e^x$ to $y=3e^(6x) -1$ .
- 1. 1. Horizontal dilation, 2. Vertical dilation, 3. Vertical translation
- 2. 1. Horizontal translation, 2. Vertical dilation, 3. Vertical translation
- 3. 1. Horizontal dilation, 2. Vertical translation, 3. Vertical translation
- 4. 1. Vertical translation, 2. Horizontal dilation, 3. Vertical dilation
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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 $y=f(x)+color(purple)(c)$ where $color(purple)(c)$ is an upward movement along the $y$ -axis.

The application of the horizontal dilation ( $color(blue)(text{factor})$ ) on a $x$ variable is $color(blue)(text{factor})=1/color(green)(a)$ and we can apply $x/color(blue)(text{factor})$ .

A standard function form for a vertical dilation factor is $y=color(red)(k)f(x)$ where $color(red)(k)$ is the vertical dilation factor.

To find the correct order so we can go from $y=e^x$ to $y=3e^(6x) -1$ , lets try a horizontal dilation.

The first transformation to be applied to $y=e^x$ will be the horizontal dilation $color(blue)(text{factor})=1/color(green)(a)$ where $color(green)(a=6)$ .
Notice $color(blue)(\text(factor)=1/6)$ .

This is from the given equation $y=3e^(color(green)(6)x)-1$ .

This is in the form $y=e^(color(green)(a)x)$ where $a=6$ , so:

$y=$ $e^(color(green)(6)x)$

Alternatively, we could use $x/color(blue)(text{factor})$ which equals $e^(x/(color(blue)(1/6)))$ which equals: $y=e^(color(green)(6)x)$ .

The second transformation to be applied on $y=e^(6x)$ will be the vertical dilation factor $color(red)(k)$ where $color(red)(k=3)$ .

$y=$ $color(red)(3)e^(6x)$ Apply the vertical dilation factor $color(red)(k=3)$ . Remember $y=color(red)(k)f(x)$ .

The third transformation to be applied on $y=3e^(6x)$ will be the vertical translation $y=f(x)+color(purple)(c)$ where $color(purple)(c=-1)$ .

$y=$ $3e^(6x)color(purple)(-1)$ Apply the vertical translation $color(purple)(c=-1)$ .

Yes, this is the correct order! It matches the original function.

1. Horizontal dilation, 2. Vertical dilation, 3. Vertical translation
Question 6 of 8

6. Question
Describe the order of the transformations when transforming $y=logx$ to $y=7log(x+5)-4$ .
- 1. 1. Vertical translation, 2. Vertical dilation, 3. Horizontal translation
- 2. 1. Vertical translation, 2. Vertical dilation, 3. Vertical translation
- 3. 1. Horizontal translation, 2. Vertical dilation, 3. Vertical translation
- 4. 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 $y=f(x)+color(purple)(c)$ where $color(purple)(+c)$ is an upward movement along the $y$ -axis.

A standard function form for a horizontal translation is $y=f(x+color(blue)(h))$ where $color(blue)(h)$ is a shift to the left along the $x$ -axis.

A standard function form for a vertical dilation factor is $y=color(red)(k)f(x)$ where $color(red)(k)$ 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 $color(blue)(h)$ where $color(blue)(h=+5)$ .

$y=$ $log(xcolor(blue)(+5))$ Apply the horizontal translation $color(blue)(h=+5)$ . Remember $y=f(x+color(blue)(h))$ .

The second transformation to be applied on $y=log(x+5)$ will be the vertical dilation factor $color(red)(k)$ where $color(red)(k=7)$ .

$y=$ $color(red)(7)log(x+5)$ Apply the vertical dilation factor $color(red)(k=7)$ . Remember $y=color(red)(k)f(x)$ .

The third transformation to be applied on $y=7log(x+5)$ will be the vertical translation $y=f(x)+color(purple)(c)$ where $color(purple)(c=-4)$ .

$y=$ $7log(x+5)color(purple)(-4)$ Apply the vertical translation $color(purple)(c=-4)$ . Remember $y=f(x)+color(purple)(c)$ .

Yes, this is the correct order! It matches the original function.

1. Horizontal translation, 2. Vertical dilation, 3. Vertical translation
Question 7 of 8

7. Question
Describe the order of the transformations when transforming $y=|x|$ to $y=|-4(x+2)|+6$ .
- 1. 1. Reflection about the $y$ -axis 2. Horizontal dilation, 3. Horizontal translation, 4. Vertical translation
- 2. 1. Reflection about the $y$ -axis 2. Vertical dilation, 3. Vertical translation, 4. Horizontal translation
- 3. 1. Reflection about the $x$ -axis 2. Horizontal dilation, 3. Horizontal translation, 4. Vertical translation
- 4. 1. Vertical translation 2. Horizontal dilation, 3. Horizontal translation, 4. Reflection about the $y$ -axis
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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 $y=f(x)+color(purple)(c)$ where $color(purple)(+c)$ is an upward movement along the $y$ -axis.

A standard function form for a horizontal translation is $y=f(x+color(red)(h))$ where $color(red)(+h)$ is a left movement along the $x$ -axis.

The application of the horizontal dilation ( $color(blue)(text{factor})$ ) on a $x$ variable is $color(blue)(text{factor})=1/color(green)(a)$ or we can use $x/color(blue)(text{factor})$ .

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 $color(blue)(text{factor})=1/color(green)(a)$ .

This is in the form $y=|ax|$ . Where $a=4$

Therefore $y=|-4x|$

Alternatively, we can use $color(blue)(\text(factor)=1/a)$ which equals $1/4$ .
Therefore horizontal $color(blue)(\text(factor)=1/4)$

Now using $x/(color(blue)(\text(factor)))=|(x)/(color(blue)(1/4))|$

Therefore $y=$ $|-color(green)(4)x|$

The third transformation to be applied to $y=|-4x|$ will be the horizontal translation $color(red)(h)$ where $color(red)(h=+2)$ (where it is $2$ units to the left).

$y=$ $|-4(xcolor(red)(+2))|$ Apply the horizontal translation $color(red)(h=+2)$ . Remember $y=f(x+color(red)(h))$ .

The fourth transformation to be applied on $y=|-4(x+2)|$ will be the vertical translation $y=f(x)+color(purple)(c)$ where $color(purple)(c=+6)$ .

$y=$ $|-4(x+2)|+color(purple)(6)$

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
Question 8 of 8

8. Question
Describe the order of the transformations when transforming $y=x^3$ to $y=8(3x-9)^3 +2$ .
- 1. 1. Horizontal translation, 2. Horizontal dilation, 3. Vertical dilation, 4 Vertical translation
- 2. 1. Vertical dilation, 2. Horizontal translation, 3. Horizontal dilation, 4 Vertical translation
- 3. 1. Horizontal dilation, 2. Horizontal translation, 3. Vertical dilation, 4. Vertical translation
- 4. 1. Vertical dilation, 2. Vertical translation, 3. Horizontal dilation, 4. Horizontal 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 translation

A standard function form for a vertical translation is $y=f(x)+color(purple)(c)$ where $color(purple)(+c)$ is an upward movement along the $y$ -axis.

A standard function form for a horizontal translation is $y=f(x+color(red)(h))$ where $color(red)(+h)$ is a shift to the left along the $x$ -axis.

A standard function form for a vertical dilation factor is $y=color(orange)(k)f(x)$ where $color(orange)(k)$ is the vertical dilation factor.

The application of the horizontal dilation $color(blue)(text{factor})$ on a $x$ variable is $color(blue)(text{factor})=1/color(green)(a)$ or we can apply $x/color(blue)(text{factor})$ .

The universal formula form with two translations and two dilations factors in one equation is $y = color(orange)(k)f(color(green)(a)(x+color(red)(b)))+color(purple)(c)$ , where $color(red)(b)$ is also $color(red)(h)$ in the horizontal translation.
$color(orange)(k)$ is the vertical dilation factor. $color(red)(+b)$ is shift left. $color(purple)(+c)$ is shift up. Horizontal factor is $1/color(green)(a).$

To find the correct order we will take a guess to go from $y=x^3$ 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 $y= color(orange)(k)f(color(green)(a)(x+color(red)(b)))+color(purple)(c)$ .

The first transformation to be applied to $y=x^3$ will be the horizontal dilation $color(blue)(text{factor})=1/color(green)(a)$ where $color(green)(a=3)$ (from universal formula).

$y=$ $(color(green)(3)x)^3$ Apply the horizontal parameter $color(green)(a=3)$ . Remember $color(blue)(text{factor})=1/color(green)(a)$ and $x/color(blue)(text{factor})$ .

The second transformation to be applied to $y=(3x)^3$ will be the horizontal translation $color(red)(h)$ where $color(red)(h=-3)$ . Simply replace $x$ for $x-3$ .

$y=$ $(3(xcolor(red)( – 3)))^3$ Apply the horizontal translation $color(red)(h=-3)$ . Remember $y=f(x+color(red)(h))$ .

The third transformation to be applied on $y=(3(x-3))^3$ will be the vertical dilation factor $color(orange)(k)$ where $color(orange)(k=8)$ .

$y=$ $color(orange)(8)(3(x-3))^3$ Apply the vertical dilation factor $color(orange)(k=8)$ . Remember $y=color(orange)(k)f(x)$ .

The fourth transformation to be applied on $y=8(3(x-3))^3$ will be the vertical translation $y=f(x)+color(purple)(c)$ where $color(purple)(c=+2)$ .

$y=$ $8(3(x-3))^3+color(purple)(2)$ Apply the vertical translation $color(purple)(c=2)$ . Remember $y=f(x)+color(purple)(c)$ .

Yes, this is the correct order! It matches the original function.

1. Horizontal dilation, 2. Horizontal translation, 3. Vertical dilation, 4. Vertical translation

$y=$		$(x/color(blue)(5))+3)^2$	Apply the horizontal dilation $color(blue)(\text(factor)=5)$ . We apply $x/color(blue)(text{factor})$ by taking the $x$ and dividing it by $5$ .
$y=$		$(1/5x+3)^2$

Combinations of Transformations: Order

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Quizzes

$y$	$=$	$x^3$	Then we use $x/\text(factor)$ where we divide $x$ by $color(blue)(1/5)$
$y$	$=$	$(x/(color(blue)(1/5)))^3$	Simplify.
$y$	$=$	$(5x)^3$

$y=$	$ln x$	We will apply $x/color(blue)(text{factor})$ , take the $x$ and divide by $1/4$ .
$y=$	$ln (x/(1/4))$	Simplify
$y=$	$ln 4x$