Combinations of Transformations: Find Equation 2

Question 1 of 6

Find the transformed version of $y = \frac{1}{x}$ $y=1/x$ when a horizontal translation of $3$ $3$ units left, a vertical dilation of $2$ $2$ , and a vertical translation of $4$ $4$ units down are applied.

1. $y = \frac{2}{x + 3} - 4$ $y=2/(x+3)-4$
2. $y = \frac{2}{x + 3} + 4$ $y=2/(x+3)+4$
3. $y = \frac{2}{x - 3} + 4$ $y=2/(x-3)+4$
4. $y = \frac{2}{x - 3} - 4$ $y=2/(x-3)-4$

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A standard function form for a horizontal translation is

y = f (x + h)

$y=f(x+color(red)(h))$ where

+ h

$color(red)(+h)$ is a shift to the left movement along the

x

$x$ -axis.

A standard function form for a vertical translation is

y = f (x) + c

$y=f(x)+color(purple)(c)$ where

+ c

$color(purple)(+c)$ is an upward movement along the

y

$y$ -axis.

A standard function form for a vertical dilation is

k f (x)

$color(blue)(k)f(x)$ where

k

$color(blue)(k)$ is the vertical dilation factor.

To transform

y = \frac{1}{x}

$y=1/x$ with a horizontal translation of

3

$color(red)(3)$ units left, a vertical dilation of

2

$color(blue)(2)$ , and a vertical translation of

4

$color(purple)(4)$ units down, first apply a horizontal translation moving to the left (negative) which means that

+ h

$color(red)(+h)$ is a positive. Do this by using

y = f (x + h)

$y=f(x+color(red)(h))$ and

h = 3

$color(red)(h=3)$ .

$y =$ $y=$	$\frac{1}{x}$ $1/x$	Apply the horizontal translation of $h = 3$ $color(red)(h=3)$ . Remember $y = f (x + h)$ $y=f(x+color(red)(h))$ .
$=$ $=$	$\frac{1}{x + 3}$ $1/(xcolor(red)(+3))$	Simplify
$=$ $=$	$\frac{1}{x + 3}$ $1/(x+3)$

Now apply the vertical dilation of

k = 2

$color(blue)(k=2)$ . Use

k f (x)

$color(blue)(k)f(x)$ .

$y =$ $y=$	$\frac{1}{x + 3}$ $1/(x+3)$	Apply the vertical dilation of $k = 2$ $color(blue)(k=2)$ . Use $k f (x)$ $color(blue)(k)f(x)$ .
$=$ $=$	$2 \times (\frac{1}{x + 3})$ $color(blue)(2)times(1/(x+3))$	Simplify
$=$ $=$	$\frac{2}{x + 3}$ $2/(x+3)$

Finally apply the vertical translation of

4

$color(purple)(4)$ units down. Use

y = f (x) + c

$y=f(x)+color(purple)(c)$ where

c

$color(purple)(c)$ is an upward movement along the

y

$y$ -axis. In this case

c = - 4

$color(purple)(c=-4)$ because we are going downward.

$y =$ $y=$	$\frac{2}{x + 3}$ $2/(x+3)$	Apply the vertical translation of $4$ $color(purple)(4)$ units down. This means $c = - 4$ $color(purple)(c=-4)$ .
$=$ $=$	$\frac{2}{x + 3} - 4$ $2/(x+3)color(purple)(-4)$	Simplify
$=$ $=$	$\frac{2}{x + 3} - 4$ $2/(x+3)-4$

y = \frac{2}{x + 3} - 4

$y=2/(x+3)-4$

Question 2 of 6

Find the equation when $y = x^{3}$ $y=x^3$ is transformed with a horizontal dilation factor of $2$ $2$ then a vertical translation of $3$ $3$ units up.

1. $y = \frac{x^{3}}{8} + 3$ $y=x^3/8 +3$
2. $y = \frac{1}{2} {(x + 3)}^{3}$ $y=1/2(x+3)^3$
3. $y = \frac{x^{3}}{2} + 3$ $y=x^3/2 +3$
4. $y = \frac{{(x + 3)}^{3}}{2}$ $y=(x+3)^3/2$

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The application of the horizontal dilation (

factor

$color(blue)(text{factor})$ ) on a

x

$x$ variable is

\frac{x}{factor}

$x/color(blue)(text{factor})$ .

A standard function form for a vertical translation is

y = f (x) + c

$y=f(x)+color(purple)(c)$ where

c

$color(purple)(c)$ is an upward movement along the

y

$y$ -axis.

To determine the transformed equation, start by applying the horizontal dilation factor of

2

$color(blue)(2)$ first. Do this by using

\frac{x}{factor}

$x/color(blue)(text{factor})$ and

factor = 2

$color(blue)(text{factor})color(blue)(=2)$ .

$y =$ $y=$	$x^{3}$ $x^3$	Apply the horizontal dilation factor of $2$ $color(blue)(2)$ . Remember $\frac{x}{factor}$ $x/color(blue)(text{factor})$ .
$=$ $=$	${(\frac{x}{2})}^{3}$ $(x/{color(blue)(2)})^3$	Simplify
$=$ $=$	${(\frac{x}{2})}^{3}$ $(x/2)^3$	Apply the vertical translation of $3$ $color(purple)(3)$ units up. Use $y = f (x) + c$ $y=f(x)+color(purple)(c)$ where $c$ $color(purple)(c)$ is an upward movement along the $y$ $y$ -axis. This means $c = 3$ $color(purple)(c=3)$ .
$=$ $=$	${(\frac{x}{2})}^{3} + 3$ $(x/2)^3 color(purple)(+3)$	Simplify
$=$ $=$	${(\frac{x}{2})}^{3} + 3$ $(x/2)^3 +3$	Simplify.
$y =$ $y=$	$\frac{x^{3}}{8} + 3$ $x^3/8 +3$	Simplify.

y = \frac{x^{3}}{8} + 3

$y=x^3/8 +3$

Question 3 of 6

Find the equation when $y = \sqrt{x}$ $y=sqrt(x)$ is reflected about the $y$ $y$ -axis then transformed with a vertical dilation factor of $2$ $2$ , then a horizontal dilation factor of $\frac{1}{3}$ $1/3$ .

1. $y = 2 \sqrt{- \frac{x}{3}}$ $y=2sqrt(-x/3)$
2. $y = 2 \sqrt{- 3 x}$ $y=2sqrt(-3x)$
3. $y = 2 \sqrt{3 x}$ $y=2sqrt(3x)$
4. $y = 2 \sqrt{\frac{x}{3}}$ $y=2sqrt(x/3)$

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Reflections about the

y

$y$ -axis have the property where we replace

x \to - x

$x\rightarrow-x$ .
A standard function form for a vertical dilation is

k f (x)

$color(blue)(k)f(x)$ where

k

$color(blue)(k)$ is the vertical dilation.
The application of the horizontal dilation (

factor

$color(green)(text{factor})$ ) on a

x

$x$ variable is

\frac{x}{factor}

$x/color(green)(text{factor})$ .

To determine the transformed equation, start by applying the reflection around the

y

$y$ -axis by replacing

x \to - x

$x\rightarrow-x$ .

$y =$ $y=$	$\sqrt{- x}$ $sqrt(-x)$	Replace $x$ $x$ by $- x$ $-x$ .
$=$ $=$	$2 \sqrt{- x}$ $color(blue)(2)sqrt(-x)$	Apply the vertical dilation of $k = 2$ $color(blue)(k=2)$ . Use $k f (x)$ $color(blue)(k)f(x)$ .
$=$ $=$	$2 \sqrt{- x}$ $2sqrt(-x)$	Apply the horizontal dilation factor of $\frac{1}{3}$ $color(green)(1/3)$ . Remember $\frac{x}{factor}$ $x/color(green)(text{factor})$
$=$ $=$	$2 \sqrt{- \frac{x}{\frac{1}{3}}}$ $2sqrt(-x/color(green)(1/3))$	Simplify
$y =$ $y=$	$2 \sqrt{- 3 x}$ $2sqrt(-3x)$

y = 2 \sqrt{- 3 x}

$y=2sqrt(-3x)$

Question 4 of 6

Find the transformed equation when the original function $y = \log x$ $y=logx$ is translated horizontally by $2$ $2$ units to the right, then transformed with the horizontal dilation factor of $3$ $3$ .

1. $y = \log (\frac{1}{3} x) - 2$ $y=log(1/3x)-2$
2. $y = \log (\frac{1}{3} x - 2)$ $y=log(1/3x - 2)$
3. $y = \frac{1}{3} \log (1 x - 2)$ $y=1/3log(1x - 2)$
4. $y = \log (3 x - 2)$ $y=log(3x - 2)$

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A standard function form for a horizontal translation is

y = f (x + h)

$y=f(x+color(red)(h))$ where

+ h

$color(red)(+h)$ is a shift to the left movement along the

x

$x$ -axis.

The application of the horizontal dilation (

factor

$color(blue)(text{factor})$ ) on a

x

$x$ variable is

\frac{x}{factor}

$x/color(blue)(text{factor})$ .

To determine the transformed equation, we start by applying the horizontal translation of

2

$color(red)(2)$ units right. Use

y = f (x + h)

$y=f(x+color(red)(h))$ where

h

$color(red)(h)$ is a shift to the left along the

x

$x$ -axis. This means

h = - 2

$color(red)(h=-2)$ .

$y =$ $y=$	$\log x$ $logx$	Apply the horizontal translation of $2$ $color(red)(2)$ units right. Use $y = f (x - 2)$ $y=f(xcolor(red)(-2))$ .
$=$ $=$	$\log (x - 2)$ $log(xcolor(red)(-2))$	Simplify
$=$ $=$	$\log (x - 2)$ $log(x-2)$	Apply the horizontal dilation factor of $3$ $color(blue)(3)$ . Remember $\frac{x}{factor}$ $x/color(blue)(text{factor})$ .
$=$ $=$	$\log ((\frac{x}{3}) - 2)$ $log((x/color(blue)(3)) -2)$	Simplify
$y =$ $y=$	$\log (\frac{1}{3} x - 2)$ $log(1/3x – 2)$

y = \log (\frac{1}{3} x - 2)

$y=log(1/3x – 2)$

Question 5 of 6

Find the transformed version of $y = 3^{x}$ $y=3^x$ when a vertical translation of $2$ $2$ units up, a horizontal translation of $2$ $2$ units to the right, and a vertical dilation factor of $- 1$ $-1$ are applied.

1. $y = - 3^{x - 2} - 2$ $y=-3^(x-2) - 2$
2. $y = 3^{x - 2} + 2$ $y=3^(x-2) + 2$
3. $y = 3^{x + 2} + 2$ $y=3^(x+2) + 2$
4. $y = - 3^{x + 2} - 2$ $y=-3^(x+2) - 2$

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A standard function form for a horizontal translation is

y = f (x + h)

$y=f(x+color(red)(h))$ where

+ h

$color(red)(+h)$ is a shift to the left movement along the

x

$x$ -axis.

A standard function form for a vertical translation is

y = f (x) + c

$y=f(x)+color(purple)(c)$ where

+ c

$color(purple)(+c)$ is an upward movement along the

y

$y$ -axis.

A standard function form for a vertical dilation is

k f (x)

$color(blue)(k)f(x)$ where

k

$color(blue)(k)$ is the vertical dilation.

To transform

y = 3^{x}

$y=3^x$ with a vertical translation of

2

$color(purple)(2)$ units up, horizontal translation of

2

$color(red)(2)$ units right, and a vertical dilation of

- 1

$color(blue)(-1)$ , first apply a vertical translation of

2

$2$ units up. This means

c = 2

$color(purple)(c=2)$ .

$y =$ $y=$	$3^{x}$ $3^x$	Apply the vertical translation of $2$ $color(purple)(2)$ units up. Use $y = f (x) + c$ $y=f(x)+color(purple)(c)$ where $+ c$ $color(purple)(+c)$ is an upward movement along the $y$ $y$ -axis. This means $c = 2$ $color(purple)(c=2)$ .
$=$ $=$	$3^{x} + 2$ $3^x + color(purple)(2)$	Simplify
$=$ $=$	$3^{x} + 2$ $3^x + 2$

Next, remember

+ h

$color(red)(+h)$ (left) is a shift to the left along the

x

$x$ -axis. So moving to the right,

h

$color(red)(h)$ is a negative. Do this by using

y = f (x + h)

$y=f(x+color(red)(h))$ and

h = - 2

$color(red)(h=-2)$ .

$y =$ $y=$	$3^{x} + 2$ $3^x + 2$	Apply the horizontal translation of $h = - 2$ $color(red)(h=-2)$ . Remember $y = f (x + h)$ $y=f(x+color(red)(h))$ .
$=$ $=$	$3^{x - 2} + 2$ $3^(xcolor(red)(-2)) + 2$	Simplify
$=$ $=$	$3^{x - 2} + 2$ $3^(x-2) + 2$

Now apply the vertical dilation of

k = - 1

$color(blue)(k=-1)$ . Use

k f (x)

$color(blue)(k)f(x)$ .

$y =$ $y=$	$3^{x - 2} + 2$ $3^(x-2) + 2$	Apply the vertical dilation of $k = - 1$ $color(blue)(k=-1)$ . Use $k f (x)$ $color(blue)(k)f(x)$ .
$=$ $=$	$- 1 [3^{x - 2} + 2]$ $color(blue)(-1)[3^(x-2) + 2]$	Simplify
$=$ $=$	$- 3^{x - 2} - 2$ $-3^(x-2) – 2$

y = - 3^{x - 2} - 2

$y=-3^(x-2) – 2$

Question 6 of 6

Find the equation when $y = \sqrt{x}$ $y=sqrt(x)$ is transformed with a vertical translation of $3$ $3$ units up, a horizontal dilation factor of $2$ $2$ , a horizontal translation of $1$ $1$ unit up, and a reflection about the $x$ $x$ -axis.

1. $y = \sqrt{\frac{1}{2} (x - 1)} + 3$ $y=sqrt(1/2(x-1))+3$
2. $y = - \sqrt{\frac{1}{2} (x + 1)} - 3$ $y=-sqrt(1/2(x+1))-3$
3. $y = - \sqrt{\frac{1}{2} (x - 1)} - 3$ $y=-sqrt(1/2(x-1))-3$
4. $y = - \sqrt{\frac{1}{2} (x - 1)} + 3$ $y=-sqrt(1/2(x-1))+3$

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A standard function form for a horizontal translation is

y = f (x + h)

$y=f(x+color(red)(h))$ where

+ h

$color(red)(+h)$ is a shift to the left movement along the

x

$x$ -axis.

A standard function form for a vertical translation is

y = f (x) + c

$y=f(x)+color(purple)(c)$ where

+ c

$color(purple)(+c)$ is an upward movement along the

y

$y$ -axis.

The application of the horizontal dilation

factor

$color(blue)(text{factor})$ on a

x

$x$ variable is

\frac{x}{factor}

$x/color(blue)(text{factor})$ .

Reflections about the

x

$x$ -axis have the property where we replace

y \to - y

$y\rightarrow-y$ .

To determine the transformed equation, start by applying the vertical translation of

3

$color(purple)(3)$ units up. Use

y = f (x) + c

$y=f(x)+color(purple)(c)$ where

+ c

$color(purple)(+c)$ is an upward movement along the

y

$y$ -axis. This means

c = 3

$color(purple)(c=3)$ .

$y =$ $y=$	$\sqrt{x} + 3$ $sqrt(x)color(purple)(+3)$	Apply the vertical translation $c = 3$ $color(purple)(c=3)$ .
$=$ $=$	$\sqrt{\frac{x}{2}} + 3$ $sqrt(x/color(blue)(2))+3$	Apply the horizontal dilation factor of $2$ $color(blue)(2)$ . Remember $\frac{x}{factor}$ $x/color(blue)(text{factor})$ .
$=$ $=$	$\sqrt{\frac{1}{2} (x - 1)} + 3$ $sqrt(1/2(xcolor(red)(-1)))+3$	Apply the horizontal translation $- 1$ $color(red)(-1)$ unit right.
$=$ $=$	$- [\sqrt{\frac{1}{2} (x - 1)} + 3]$ $-[sqrt(1/2(x-1))+3]$	Reflecting about the $x$ $x$ -axis. Replace $y$ $y$ for $- y$ $-y$ .
$y =$ $y=$	$- \sqrt{\frac{1}{2} (x - 1)} - 3$ $-sqrt(1/2(x-1))-3$

y = - \sqrt{\frac{1}{2} (x - 1)} - 3

$y=-sqrt(1/2(x-1))-3$

Combinations of Transformations: Find Equation 2

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