Horizontal Dilations – Scale Factor

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Question 1 of 7

1. Question

Apply the horizontal dilation (stretch/shrink) scale factor of `4` to the function `y=2^x`.

`y=2^(x/4)`
`y=2^(4x)`
`y=2^(2x)`
`y=2^(x/2)`

Correct

Great Work!

Incorrect

A Dilation is to stretch or to shrink the shape of a curve.
Horizontal dilations (stretch/shrink) of a function are written in the form `y=f(ax)`. To find `color(blue)(a)`, use the formula `color(blue)(a)=1/(color(red)(\text(Factor)))` or `color(red)(\text(Factor))=1/color(blue)(a)`.

Alternatively, we can take the `x` component of the function and divide it by the horizontal dilation factor: `\ \ \ \ \ \ x/color(red)(\text(Factor))`

To obtain the equation of the function after the horizontal dilation (stretch/shrink), first solve for `color(blue)(a)` where the `color(red)(\text(Factor)=4)`. Factor of `4` means it is stretched `4` times as wide horizontally. Use the formula `color(blue)(a)=1/(color(red)(\text(Factor)))`.

This gives `color(blue)(a)=1/(color(red)(\text(Factor)))=1/color(red)(4)`.

`y=`		`2^(color(blue)(1/4)\timesx)`	Write the new equation using `y=2^(color(blue)(a)x)` and `color(blue)(a=1/4)`. Remember horizontal dilations of a function are written in the form `y=f(color(blue)(a)x)`.
`y=`		`2^(x/4)`

`y=2^(x/4)`

Question 2 of 7

2. Question

Apply the horizontal dilation (stretch/shrink) scale factor of `1/2` to the function `y=sqrt(x)`.

`y=sqrt(2x)`
`y=sqrt(x/2)`
`y=sqrt(4x)`
`y=sqrt(x/4)`

Correct

Great Work!

Incorrect

Alternatively, we can take the `x` component of the function and divide it by the horizontal dilation factor: `\ \ x/color(red)(\text(Factor))`

To obtain the equation of the function after the horizontal dilation (stretch/shrink), first solve for `color(blue)(a)` where the `color(red)(\text(Factor)=1/2)`. Factor of `1/2` means it is compressed half as wide horizontally. Use the formula `color(blue)(a)=1/(color(red)(\text(Factor)))`.

This gives `color(blue)(a)=1/(color(red)(\text(Factor)))=1/(color(red)(1/2))=2`.

`y=`		`sqrt(color(blue)(2)\timesx)`	Write the new equation using `y=sqrt(color(blue)(a)x)` and `color(blue)(a=2)`. Remember horizontal dilations of a function are written in the form `y=f(color(blue)(a)x)`.
`y=`		`sqrt(2x)`

`y=sqrt(2x)`

Question 3 of 7

3. Question

Apply the horizontal dilation (stretch/shrink) scale factor of `1/2` to the function `y=2^x`.

`y=2^(2x)`
`y=2^(x/2)`
`y=2 \times 2^(x)`
`y=(2^(x))/2`

Correct

Great Work!

Incorrect

Horizontal dilations of a function are written in the form `y=f(ax)`. To find `color(blue)(a)`, use the formula `color(blue)(a)=1/(color(red)(\text(Factor)))` or `color(red)(\text(Factor))=1/color(blue)(a)`.

Alternatively, we can take the `x` component of the function and divide it by the horizontal dilation factor: `\ \ x/color(red)(\text(Factor))`

To obtain the equation of the function after the horizontal dilation, first solve for `color(blue)(a)` where the `color(red)(\text(Factor)=1/2)`. Factor of `1/2` means it is compressed half as wide horizontally. Use the formula `color(blue)(a)=1/(color(red)(\text(Factor)))`.

This gives `color(blue)(a)=1/(color(red)(\text(Factor)))=1/(color(red)(1/2))=2`.

`y=`		`2^(color(blue)(2)\timesx)`	Write the new equation using `y=2^(color(blue)(a)x)` and `color(blue)(a=2)`. Remember horizontal dilations of a function are written in the form `y=f(color(blue)(a)x)`.
`y=`		`2^(2x)`

`y=2^(2x)`

Question 4 of 7

4. Question

Apply the horizontal dilation (stretch/shrink) scale factor of `3` to the function `y=x^3`.

`y=(x/3)^3`
`y=(x^3)/3`
`y=(3x)^3`
`y=(x+3)^3`

Correct

Great Work!

Incorrect

Alternatively, we can take the `x` component of the function and divide it by the horizontal dilation factor: `\ \ x/color(red)(\text(Factor))`

To obtain the equation of the function after the horizontal dilation (stretch/shrink), first solve for `color(blue)(a)` where the `color(red)(\text(Factor)=3)`. Factor of `3` means it is stretched `3` times as wide horizontally. Use the formula `color(blue)(a)=1/(color(red)(\text(Factor)))`.

This gives `color(blue)(a)=1/(color(red)(\text(Factor)))=1/(color(red)(3)`.

`y=`		`(color(blue)(1/3)\timesx)^3`	Write the new equation using `y=(color(blue)(a)x)^3` and `color(blue)(a=1/3)`. Remember horizontal dilations of a function are written in the form `y=f(color(blue)(a)x)`.
`y=`		`(x/3)^3`

`y=(x/3)^3`

Question 5 of 7

5. Question

Apply the horizontal dilation (stretch/shrink) scale factor of `6` to the function `y=log(x)`.

`y=log(x/6)`
`y=log(6x)`
`y=6 \times log(x)`
`y=log(x)/6`

Correct

Great Work!

Incorrect

Alternatively, we can take the `x` component of the function and divide it by the horizontal dilation factor: `\ \ x/color(red)(\text(Factor))`

To obtain the equation of the function after the horizontal dilation (stretch/shrink), first solve for `color(blue)(a)` where the `color(red)(\text(Factor)=6)`. Factor of `6` means it is stretched `6` times as wide horizontally. Use the formula `color(blue)(a)=1/(color(red)(\text(Factor)))`.

This gives `color(blue)(a)=1/(color(red)(\text(Factor)))=1/color(red)(6)`.

`y=`		`log(color(blue)(1/6 )\timesx)`	Write the new equation using `y=log(color(blue)(a)x)` and `color(blue)(a=1/6)`. Remember horizontal dilations of a function are written in the form `y=f(color(blue)(a)x)`.
`y=`		`log(x/6)`

`y=log(x/6)`

Question 6 of 7

6. Question

Apply the horizontal dilation (stretch/shrink) scale factor of `1/8` to the function `y=log(x)`.

`y=log(8x)`
`y=log(x/8)`
`y=8log(x)`
`y=log(x)/8`

Correct

Great Work!

Incorrect

Alternatively, we can take the `x` component of the function and divide it by the horizontal dilation factor: `\ \ x/color(red)(\text(Factor))`

To obtain the equation of the function after the horizontal dilation (stretch/shrink), first solve for `color(blue)(a)` where the `color(red)(\text(Factor)=1/8)`. Factor of `1/8` means it is compressed `1/8` as wide horizontally. Use the formula `color(blue)(a)=1/(color(red)(\text(Factor)))`.

This gives `color(blue)(a)=1/(color(red)(\text(Factor)))=1/(color(red)(1/8))=8`.

`y=`		`log(color(blue)(8)\timesx)`	Write the new equation using `y=log(color(blue)(a)x)` and `color(blue)(a=8)`. Remember horizontal dilations of a function are written in the form `y=f(color(blue)(a)x)`.
`y=`		`log(8x)`

`y=log(8x)`

Question 7 of 7

7. Question

Apply the horizontal dilation (stretch/shrink) scale factor of `4` to the function `y=x^2`.

`y=(x/4)^2`
`y=(4x)^2`
`y=(x)^8`
`y=(x)^(1/2)`

Correct

Great Work!

Incorrect

Alternatively, we can take the `x` component of the function and divide it by the horizontal dilation factor: `\ \ x/color(red)(\text(Factor))`

This gives `color(blue)(a)=1/(color(red)(\text(Factor)))=1/color(red)(4)`.

`y=`		`(color(blue)(1/4)\timesx)^2`	Write the new equation using `y=(color(blue)(a)x)^2` and `color(blue)(a=1/4)`. Remember horizontal dilations of a function are written in the form `y=f(color(blue)(a)x)`.
`y=`		`(x/4)^2`

`y=(x/4)^2`

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