Horizontal and Vertical Dilations (Stretch/Shrink) 3

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Question 1 of 6

1. Question
Find the transformed function from the original function $y = x^{2}$ $y=x^2$ based on the given dilation (stretch/shrink) and scale factor.

i. Horizontally by $\frac{1}{5}$ $1/5$

ii. Vertically by $25$ $25$
- 1. i. $y = 25 x^{2}$ $y=25x^2$ and ii. $y = 25 x^{2}$ $y=25x^2$
- 2. i. $y = 25 x^{2}$ $y=25x^2$ and ii. $y = 5 x^{2}$ $y=5x^2$
- 3. i. $y = 5 x^{2}$ $y=5x^2$ and ii. $y = 25 x^{2}$ $y=25x^2$
- 4. i. $y = 5 x^{2}$ $y=5x^2$ and ii. $y = 5 x^{2}$ $y=5x^2$
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Vertical dilation takes the form $y = k f (x)$ $y=bb(k)f(x)$ where $k$ $bb(k)$ is the vertical scale factor.

Horizontal dilation takes the form $y = f (a x)$ $y=f(color(green)(a)x)$ where the scale factor can be found from $\frac{x}{Factor}$ $x/\text(Factor)$ or $Factor = \frac{1}{a}$ $\text(Factor) =1/color(green)(a)$ .

i. Since the dilation (stretch/shrink) is horizontal with a factor of $\frac{1}{5}$ $1/5$ , we follow the horizontal dilation form $y = f (a x)$ $y=f(color(green)(a)x)$ .

Calculate the horizontal scale factor by using $a = \frac{1}{Factor}$ $color(green)(a) =1/\text(Factor)$ and $Factor = \frac{1}{5}$ $\text(Factor)=1/5$ .

$a =$ $color(green)(a) =$ $\frac{1}{\frac{1}{5}}$ $1/(1/5)$ Simplify

$a =$ $color(green)(a)=$ $5$ $5$

The original function $y = x^{2}$ $y=x^2$ will become $y = {(5 x)}^{2}$ $y=(5x)^2$ or simply $y = 25 x^{2}$ $y=25x^2$ .

ii. Since the dilation is vertical with a factor of $25$ $25$ , we follow the vertical dilation form $y = (k) f (x)$ $y=(k)f(x)$ . Remember that $k$ $k$ is the scale factor.

So, we can say the $k = 25$ $k=25$ .

From there, the original function $y = x^{2}$ $y=x^2$ will become $y = 25 x^{2}$ $y=25x^2$ .

i. $y = 25 x^{2}$ $y=25x^2$ and ii. $y = 25 x^{2}$ $y=25x^2$
Question 2 of 6

2. Question
Find the transformed function from the original function $y = x^{3} + 4 x$ $y=x^3 +4x$ based on the given dilation (stretch/shrink) and scale factor.

i. Horizontally by $\frac{1}{3}$ $1/3$

ii. Vertically by $3$ $3$
- 1. i. $y = 3 x^{3} + 4 x$ $y=3x^3 + 4x$ and ii. $y = {(3 x)}^{3} + 4 x$ $y=(3x)^3 + 4x$
- 2. i. $y = 3 x^{3} + 12 x$ $y=3x^3 + 12x$ and ii. $y = {(3 x)}^{3} + 12 x$ $y=(3x)^3 + 12x$
- 3. i. $y = {(3 x)}^{3} + 4 x$ $y=(3x)^3 + 4x$ and ii. $y = 3 x^{3} + 4 x$ $y=3x^3 + 4x$
- 4. i. $y = {(3 x)}^{3} + 12 x$ $y=(3x)^3 + 12x$ and ii. $y = 3 x^{3} + 12 x$ $y=3x^3 + 12x$
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Vertical dilation takes the form $y = k f (x)$ $y=bb(k)f(x)$ where $k$ $bb(k)$ is the vertical scale factor.

Horizontal dilation takes the form $y = f (a x)$ $y=f(color(green)(a)x)$ where the scale factor can be found from $\frac{x}{Factor}$ $x/\text(Factor)$ or $Factor = \frac{1}{a}$ $\text(Factor) =1/color(green)(a)$ .

i. Since the dilation (stretch/shrink) is horizontal with a factor of $\frac{1}{3}$ $1/3$ , we follow the horizontal dilation form $y = {f (a x)}^{2}$ $y=f(color(green)(a)x)^2$ .

Calculate the horizontal scale factor by using $a = \frac{1}{Factor}$ $color(green)(a) =1/\text(Factor)$ and $Factor = \frac{1}{3}$ $\text(Factor)=1/3$ .

$a =$ $color(green)(a) =$ $\frac{1}{\frac{1}{3}}$ $1/(1/3)$ Simplify

$a =$ $color(green)(a)=$ $3$ $3$

The original function $y = x^{3} + 4 x$ $y=x^3 + 4x$ will become $y = {(3 x)}^{3} + 4 (3 x)$ $y=(3x)^3 + 4(3x)$ or simply $y = {(3 x)}^{3} + 12 x$ $y=(3x)^3 + 12x$ .

ii. Since the dilation is vertical with a factor of $3$ $3$ , we follow the vertical dilation form $y = (k) f (x)$ $y=(k)f(x)$ . Remember that $k$ $k$ is the scale factor.

So, we can say the $k = 3$ $k=3$ .

From there, the original function $y = x^{3} + 4 x$ $y=x^3 + 4x$ will become $y = 3 (x^{3} + 4 x)$ $y=3(x^3 + 4x)$ or simply $y = 3 x^{3} + 12 x$ $y=3x^3 + 12x$ .

i. $y = {(3 x)}^{3} + 12 x$ $y=(3x)^3 + 12x$ and ii. $y = 3 x^{3} + 12 x$ $y=3x^3 + 12x$
Question 3 of 6

3. Question
What dilation (stretch/shrink) is needed to transform $y = \frac{1}{x + 6}$ $y=1/(x+6)$ to $y = \frac{1}{4 x + 6}$ $y=1/(4x+6)$ ?
- 1. Vertical dilation with a scale factor of $4$ $4$
- 2. Horizontal dilation with a scale factor of $\frac{1}{4}$ $1/4$
- 3. Vertical dilation with a scale factor of $\frac{1}{4}$ $1/4$
- 4. Horizontal dilation with a scale factor of $4$ $4$
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Vertical dilation takes the form $y = k f (x)$ $y=bb(k)f(x)$ where $k$ $bb(k)$ is the vertical scale factor.

Horizontal dilation takes the form $y = f (a x)$ $y=f(color(green)(a)x)$ where the scale factor can be found from $\frac{x}{Factor}$ $x/\text(Factor)$ or $Factor = \frac{1}{a}$ $\text(Factor) =1/color(green)(a)$ .

To find which type of dilation (stretch/shrink) is happening, compare the transformed function $y = \frac{1}{(4 x) + 6}$ $y=1/((color(green)(4)x)+6)$ to the vertical dilation form $y = k f (x)$ $y=bb(k)f(x)$ and the horizontal dilation form $y = f (a x)$ $y=f(color(green)(a)x)$ .

The transformed function $y = \frac{1}{(4 x) + 6}$ $y=1/((color(green)(4)x)+6)$ looks like the horizontal dilation form $y = f (a x)$ $y=f(color(green)(a)x)$ . This means that $a = 4$ $color(green)(a=4)$ .

Calculate the horizontal scale factor by using $Factor = \frac{1}{a}$ $\text(Factor) =1/color(green)(a)$ and $a = 4$ $color(green)(a=4)$ .

$Factor =$ $\text(Factor) =$ $\frac{1}{4}$ $1/color(green)(4)$ Simplify

$Factor =$ $\text(Factor) =$ $\frac{1}{4}$ $1/4$

Horizontal dilation with a scale factor of $\frac{1}{4}$ $1/4$
Question 4 of 6

4. Question
What is the equation when $y = x^{2} + 5$ $y=x^2 +5$ is vertically dilated (stretch/shrink) by a factor of $- 1$ $-1$ .
- 1. $y = x^{2} - 5$ $y=x^2 -5$
- 2. $y = x^{2} + 5$ $y=x^2 +5$
- 3. $y = - x^{2} - 5$ $y=-x^2 -5$
- 4. $y = - x^{2} + 5$ $y=-x^2 +5$
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Vertical dilation takes the form $y = k f (x)$ $y=color(blue)(bb(k))f(x)$ where $k$ $color(blue)(bb(k))$ is the vertical scale factor.

Horizontal dilation takes the form $y = f (a x)$ $y=f(ax)$ where the scale factor can be found from $\frac{x}{Factor}$ $x/\text(Factor)$ or $Factor = \frac{1}{a}$ $\text(Factor) =1/a$ .

Since the dilation (stretch/shrink) is vertical with a factor of $- 1$ $-1$ , we follow the vertical dilation form $y = (k) f (x)$ $y=(k)f(x)$ . Remember that $k$ $k$ is the scale factor.

So, we can say the $k = - 1$ $k=-1$ .

From there, the original function $y = x^{2} + 5$ $y=x^2 +5$ will become $y = - 1 (x^{2} + 5)$ $y=-1(x^2 +5)$ or simply $y = - x^{2} - 5$ $y=-x^2 -5$ .

$y = - x^{2} - 5$ $y=-x^2 -5$
Question 5 of 6

5. Question
What is the equation when $x^{2} + y^{2} = 25$ $x^2 + y^2= 25$ is vertically dilated (stretch/shrink) by a factor of $\frac{1}{4}$ $1/4$ .
- 1. $x^{2} + {(16 y)}^{2} = 25$ $x^2 + (16y)^2= 25$
- 2. ${(16 x)}^{2} + y^{2} = 25$ $(16x)^2 + y^2= 25$
- 3. $16 x^{2} + y^{2} = 25$ $16x^2 + y^2= 25$
- 4. $x^{2} + 16 y^{2} = 25$ $x^2 + 16y^2= 25$
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Vertical dilation takes the form $y = k f (x)$ $y=color(blue)(bb(k))f(x)$ where $k$ $color(blue)(bb(k))$ is the vertical scale factor.

Horizontal dilation takes the form $y = f (a x)$ $y=f(ax)$ where the scale factor can be found from $\frac{x}{Factor}$ $x/\text(Factor)$ or $Factor = \frac{1}{a}$ $\text(Factor) =1/a$ .

Since the dilation (stretch/shrink) is vertical with a factor of $\frac{1}{4}$ $1/4$ , we follow the vertical dilation form $y = (k) f (x)$ $y=(k)f(x)$ . Remember that $k$ $k$ is the scale factor.

So, we can say the $k = \frac{1}{4}$ $k=1/4$ .

From there, the original function $x^{2} + {(\frac{y}{\frac{1}{4}})}^{2} = 25$ $x^2 + (y/(1/4))^2= 25$ will become $x^{2} + {(4 y)}^{2} = 25$ $x^2 + (4y)^2= 25$ or simply $x^{2} + 16 y^{2} = 25$ $x^2 + 16y^2= 25$ .

$x^{2} + 16 y^{2} = 25$ $x^2 + 16y^2= 25$
Question 6 of 6

6. Question
What dilation (stretch/shrink) is needed to transform $y = x^{3} + 6 x$ $y=x^3 + 6x$ to $y = \frac{1}{8} x^{3} + 3 x$ $y=1/8x^3 +3x$ ?
- 1. Horizontal dilation with a scale factor of $2$ $2$
- 2. Vertical dilation with a scale factor of $\frac{1}{2}$ $1/2$
- 3. Vertical dilation with a scale factor of $2$ $2$
- 4. Horizontal dilation with a scale factor of $\frac{1}{2}$ $1/2$
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Vertical dilation takes the form $y = k f (x)$ $y=bb(k)f(x)$ where $k$ $bb(k)$ is the vertical scale factor.

Horizontal dilation takes the form $y = f (a x)$ $y=f(color(green)(a)x)$ where the scale factor can be found from $\frac{x}{Factor}$ $x/\text(Factor)$ or $Factor = \frac{1}{a}$ $\text(Factor) =1/color(green)(a)$ .

To find which type of dilation (stretch/shrink) is happening, compare the transformed function $y = \frac{1}{8} x^{3} + 3 x$ $y=1/8x^3 +3x$ to the vertical dilation form $y = k f (x)$ $y=bb(k)f(x)$ and the horizontal dilation form $y = f (a x)$ $y=f(color(green)(a)x)$ .

The transformed function $y = \frac{1}{8} x^{3} + 3 x$ $y=1/8x^3 +3x$ looks like the horizontal dilation form $y = f (a x)$ $y=f(color(green)(a)x)$ . This means that $a = \frac{3}{6}$ $color(green)(a)=3/6$ or simply $a = \frac{1}{2}$ $color(green)(a)=1/2$ .

Calculate the horizontal scale factor by using $Factor = \frac{1}{a}$ $\text(Factor) =1/color(green)(a)$ and $a = \frac{1}{2}$ $color(green)(a=1/2)$ .

$Factor =$ $\text(Factor) =$ $\frac{1}{\frac{1}{2}}$ $1/color(green)(1/2)$ Simplify

$Factor =$ $\text(Factor) =$ $2$ $2$

Horizontal dilation with a scale factor of $2$ $2$

Horizontal and Vertical Dilations (Stretch/Shrink) 3

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Quizzes

$a =$ $color(green)(a) =$		$\frac{1}{\frac{1}{5}}$ $1/(1/5)$	Simplify
$a =$ $color(green)(a)=$		$5$ $5$

$a =$ $color(green)(a) =$		$\frac{1}{\frac{1}{3}}$ $1/(1/3)$	Simplify
$a =$ $color(green)(a)=$		$3$ $3$

$Factor =$ $\text(Factor) =$		$\frac{1}{4}$ $1/color(green)(4)$	Simplify
$Factor =$ $\text(Factor) =$		$\frac{1}{4}$ $1/4$

$Factor =$ $\text(Factor) =$		$\frac{1}{\frac{1}{2}}$ $1/color(green)(1/2)$	Simplify
$Factor =$ $\text(Factor) =$		$2$ $2$