Combinations of Transformations: Find Equation 3

Years

Try VividMath Premium to unlock full access

Start Free Trial

Time limit: 0

Quiz summary

0 of 6 questions completed

Questions:

Information

–

You have already completed the quiz before. Hence you can not start it again.

Quiz is loading...

You must sign in or sign up to start the quiz.

You have to finish following quiz, to start this quiz:

Lets see your results!

Points

Score

Time

Retry Quiz

Answered
Review

Question 1 of 6

1. Question

Find the transformed version of `y=x^4` when a horizontal dilation factor of `1/3` and a vertical translation of `6` units up are applied.

`y=81x^4+6`
`y=81x^4-6`
`y=(3x+6)^4`
`y=(3x-6)^4`

Correct

Great Work!

Incorrect

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

The application of the horizontal dilation `color(blue)(text{factor})` on a `x` variable is `x/color(blue)(text{factor})`.

To transform `y=x^4` with a horizontal dilation factor of `color(blue)(1/3)` and a vertical translation of `color(red)(6)` units up, start by applying the horizontal dilation factor. Do this by using `x/color(blue)(text{factor})` and `color(blue)(text{factor})color(blue)(=1/3)`.

`y=`	`x^4`	Apply the horizontal dilation factor of `color(blue)(1/3)`. Remember `x/color(blue)(text{factor})`.
`=`	`(x/{color(blue)(1/3)})^4`	Simplify
`=`	`(3x)^4`
`=`	`81x^4`

Now apply the vertical translation of `color(red)(6)` units up. Use `y=f(x)+color(red)(c)` where `color(red)(+c)` is an upward movement along the `y`-axis. This means `color(red)(c=6)`.

`y=`	`81x^4`	Apply the vertical translation of `color(red)(6)` units up. Use `y=f(x)+color(red)(c)` where `color(red)(c)` is an upward movement along the `y`-axis. This means `color(red)(c=6)`.
`=`	`81x^4color(red)(+6)`	Simplify
`=`	`81x^4+6`

`y=81x^4+6`

Question 2 of 6

2. Question

Find the equation when `y=lnx` is transformed with a horizontal dilation factor of `5` then a vertical translation of `2` units down.

`y=ln(x/5) -2`
`y=ln(5x) -2`
`y=ln(x/5)+2`
`y=ln({x+2}/5)`

Correct

Great Work!

Incorrect

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

The application of the horizontal dilation `color(blue)(text{factor})` on a `x` variable is `x/color(blue)(text{factor})`.

To determine the transformed equation, start by applying the horizontal dilation factor of `color(blue)(5)` first. Do this by using `x/color(blue)(text{factor})` and `color(blue)(text{factor})color(blue)(=5)`.

`y=`	`lnx`	Apply the horizontal dilation factor of `color(blue)(5)`. Remember `x/color(blue)(text{factor})`.
`=`	`ln(x/color(blue)(5))`	Simplify
`=`	`ln(x/5)`	Apply the vertical translation of `color(red)(2)` units down. Use `y=f(x)+color(red)(c)` where `color(red)(+c)` is an upward movement along the `y`-axis. This means `color(red)(c=-2)`.
`=`	`ln(x/5)color(purple)(-2)`	Simplify
`y=`	`ln(x/5) -2`

`y=ln(x/5) -2`

Question 3 of 6

3. Question

Find the equation when `y=logx` is transformed with a vertical dilation factor of `4` then a horizontal dilation factor of `1/2`.

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

Correct

Great Work!

Incorrect

A standard function form for a vertical dilation is `color(blue)(k)f(x)` where `color(blue)(k)` is the vertical dilation. The application of the horizontal dilation factor (`color(green)(text{factor})`) on a `x` variable is `x/color(green)(text{factor})`.

To determine the transformed equation, start by applying the vertical dilation factor of `color(blue)(4)` first.

`y=`	`logx`	Apply the vertical dilation of `color(blue)(k=4)`. Use `color(blue)(k)f(x)`.
`=`	`color(blue)(4)logx`	Apply the horizontal dilation factor of `color(green)(1/2)`. Remember `x/color(green)(text{factor})`
`=`	`4log(x/color(green)(1/2))`	Simplify
`y=`	`4log(2x)`

`y=4log(2x)`

Question 4 of 6

4. Question

Find the transformed version of `y=x^2` when a vertical translation of `2` units up, a horizontal dilation of `4` units to the left, and a horizontal translation of `1` unit right are applied.

`y=1/16(x-1)^2 + 2`
`y=1/4(x-1)^2 + 2`
`y=1/16(x+2)^2 -1`
`y=1/16(x+1)^2 - 2`

Correct

Great Work!

Incorrect

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.

The application of the horizontal dilation (`color(blue)(text{factor})`) on a `x` variable is `x/color(blue)(text{factor})`.

To transform `y=x^2` when a vertical translation of `color(purple)(2)` units up, a horizontal dilation of `color(blue)(4)` and a horizontal translation of `color(red)(1)` unit right, start by applying the vertical translation. 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)`.

`y=`	`x^2`	Apply the vertical translation of `color(purple)(c=2)`. Remember `y=f(x)+color(purple)(c)`.
`=`	`x^2 color(purple)(+2)`	Simplify
`=`	`x^2 + 2`

Now apply the horizontal dilation of `color(blue)(4)`. Do this by using `x/color(blue)(text{factor})` and `color(blue)(text{factor})color(blue)(=4)`.

`y=`	`x^2 + 2`	Apply the horizontal dilation of `color(blue)(4)`. Remember `x/color(blue)(text{factor})`.
`=`	`(x/color(blue)(4))^2 + 2`	Simplify
`=`	`x^2/16 + 2`

Finally apply the horizontal translation of `color(red)(1)` unit right. Do this by using `y=f(x+color(red)(h))` and `color(red)(h=-1)`

`y=`	`x^2/16 + 2`	Apply the horizontal translation of `color(red)(1)` unit right. Use `y=f(x+color(red)(h))`. In this case `color(red)(-h)` is a shift to the right along the `x`-axis. This means `color(red)(h=-1)`.
`=`	`(xcolor(red)(-1))^2/16 + 2`	Simplify
`=`	`1/16(x-1)^2 + 2`

`y=1/16(x-1)^2 + 2`

Question 5 of 6

5. Question

Find the equation if `x^2 + y^2 = 9` is shifted `3` units down and transformed with a vertical dilation factor of `1/3`.

`x^2 + 9(y+1)^2 =9`
`x^2 + 9(y-1)^2 =9`
`x^2 + (3y+1)^2 =9`
`x^2 + (y/3+3)^2 =9`

Correct

Great Work!

Incorrect

A standard circle equation is in the form `(x-color(red)(h))^2+(y-color(purple)(c))^2=r^2` where:

`color(red)(h)` is the `x`-coordinate of the vertex
`color(purple)(c)` is the `y`-coordinate of the vertex

Vertical Dilation factor is `color(blue)(k)`, `y=color(blue)(k)f(x) :. y/color(blue)(k)=y/color(blue)(\text(factor))=f(x)`

First we start with a vertical translation of `3` units down.

`x^2 + y^2 =`	`9`	Apply the vertical translation of `color(purple)(3)` units down. Use `color(purple)(c=-3)`.
`(x-h)^2+(y-c)^2=`	`9`
`(x-0)^2+(y-color(purple)(-3))^2=`	`9`
`x^2 + (ycolor(purple)(+3))^2 =`	`9`	Simplify
`x^2 + (y+3)^2 =`	`9`

Now apply the vertical dilation factor of `color(blue)(k=1/3)`. Use `y= color(blue)(k)f(x)`.

`y/color(blue)(k)` is the same as `y/color(blue)(\text(factor))=(y)/(color(blue)(1/3))`

`x^2 + (y+3)^2 =`	`9`	Apply the vertical dilation factor `color(blue)(k=1/3)`.
`x^2 + ((y/color(blue)(k=1/3))+3)^2 =`	`9`	Simplify
`x^2 + (3y+3)^2 =`	`9`	Factor `3`.
`x^2 + 9(y+1)^2=`	`9`

`x^2 + 9(y+1)^2 =9`

Question 6 of 6

6. Question

Find the transformed equation when the original function `y=x^2` is translated horizontally `4` units to the left, then transformed with the horizontal dilation factor of `1/4`.

`y=16(x+1)^2`
`y=2(2x+1)^2`
`y=(4x+1)^2`
`y=(x/4+2)^2`

Correct

Great Work!

Incorrect

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. The application of the horizontal dilation (`color(blue)(text{factor})`) on a `x` variable is `x/color(blue)(text{factor})`.

To determine the transformed equation, we start by applying the horizontal translation of `color(red)(4)` units left. Use `y=f(x+color(red)(h))` where `color(red)(+h)` is a shift left along the `x`-axis. This means `color(red)(h=+4)`.

`y=`	`x^2`	Apply the horizontal translation of `color(red)(+4)` units left. Use `y=f(xcolor(red)(+4))`.
`=`	`(xcolor(red)(+4))^2`	Simplify
`=`	`(x+4)^2`	Apply the horizontal dilation factor of `color(blue)(1/4)`. Remember `x/color(blue)(text{factor})`.
`=`	`(x/color(blue)(1/4) +4)^2`	Simplify
`y=`	`(4x+4)^2`	Factor out `4`.
`y=`	`16(x+1)^2`

`y=16(x+1)^2`

Quizzes

Vertical Translations 1
Vertical Translations 2
Vertical Translations from a Point
Horizontal Translations 1
Horizontal Translations 2
Horizontal Translations from a Point
Horizontal Translations from a Graph
Horizontal and Vertical Translations from a Graph
Sketch a Graph using Translations
Write the Equation from a Graph
Write the Equation from Translations
Vertical Dilations
Horizontal Dilations 1
Horizontal Dilations 2
Horizontal Dilations – Scale Factor
Horizontal and Vertical Dilations 1
Horizontal and Vertical Dilations 2
Horizontal and Vertical Dilations 3
Graphing Reflections 1
Graphing Reflections 2
Reflection with Rotation
Combinations of Transformations: Order
Combinations of Transformations: Coordinates
Combinations of Transformations: Find Equation 1
Combinations of Transformations: Find Equation 2
Combinations of Transformations: Find Equation 3