Combinations of Transformations: Find Equation 2

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Question 1 of 6

1. Question
Find the transformed version of $y=1/x$ when a horizontal translation of $3$ units left, a vertical dilation of $2$ , and a vertical translation of $4$ units down are applied.
- 1. $y=2/(x+3)-4$
- 2. $y=2/(x+3)+4$
- 3. $y=2/(x-3)+4$
- 4. $y=2/(x-3)-4$
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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 movement along the $x$ -axis.

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

To transform $y=1/x$ with a horizontal translation of $color(red)(3)$ units left, a vertical dilation of $color(blue)(2)$ , and a vertical translation of $color(purple)(4)$ units down, first apply a horizontal translation moving to the left (negative) which means that $color(red)(+h)$ is a positive. Do this by using $y=f(x+color(red)(h))$ and $color(red)(h=3)$ .

$y=$ $1/x$ Apply the horizontal translation of $color(red)(h=3)$ . Remember $y=f(x+color(red)(h))$ .

$=$ $1/(xcolor(red)(+3))$ Simplify

$=$ $1/(x+3)$

Now apply the vertical dilation of $color(blue)(k=2)$ . Use $color(blue)(k)f(x)$ .

$y=$ $1/(x+3)$ Apply the vertical dilation of $color(blue)(k=2)$ . Use $color(blue)(k)f(x)$ .

$=$ $color(blue)(2)times(1/(x+3))$ Simplify

$=$ $2/(x+3)$

Finally apply the vertical translation of $color(purple)(4)$ units down. Use $y=f(x)+color(purple)(c)$ where $color(purple)(c)$ is an upward movement along the $y$ -axis. In this case $color(purple)(c=-4)$ because we are going downward.

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

$=$ $2/(x+3)color(purple)(-4)$ Simplify

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

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

Question 2 of 6

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

1. $y=x^3/2 +3$
2. $y=x^3/8 +3$
3. $y=(x+3)^3/2$
4. $y=1/2(x+3)^3$

Question 3 of 6

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

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

Question 4 of 6

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

1. $y=1/3log(1x - 2)$
2. $y=log(3x - 2)$
3. $y=log(1/3x - 2)$
4. $y=log(1/3x)-2$

Question 5 of 6

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

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

Question 6 of 6

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

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

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

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

Combinations of Transformations: Find Equation 2

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Quizzes

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

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

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

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

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

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