Write the Equation from Translations

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Question 1 of 6

1. Question

Find the equation of the function when `y=x^2` is shifted (translated) to the right by `2` units.

`y=(x-2)^2`
`y=(x+2)^2`
`y=x^2-2`
`y=x^2+2`

Correct

Great Work!

Incorrect

Horizontal translations (`\color(royalblue)(h)`) and vertical translations (`\color(forestgreen)(c)`) of a function have the form `y=(x- color(royalblue)(h))^2+ color(forestgreen)(c)` where the point `( color(royalblue)(h), color(forestgreen)(c))` is the vertex on the function. Horizontal translations have the opposite sign as the direction of their shift.

`(-h) \ bb(rarr)` Shift Right

`(+h) \ bb(larr)` Shift Left

To obtain the equation of the function by using `y=x^2`, shift the graph to the right by `2` units. Remember squared functions have the form `y=(x- color(royalblue)(h))^2+ color(forestgreen)(c)`.

This shift is a horizontal translation (`h`) where `h=-2` inside the brackets.

`y=`		`x^2`	Write the new equation using `y=x^2` and `h=-2`. Remember horizontal translations have the opposite sign as the direction of their shift and squared functions have the form `y=(x-h)^2+c`.
`y=`		`(x-color(blue)(2))^2`

`y=(x-2)^2`

Question 2 of 6

2. Question

Find the equation of the function when `y=logx` is shifted (translated) to the left by `5` units and down by `2` units.

`y=log(x+5)-2`
`y=log(x-5)+2`
`y=log(x-5)-2`
`y=log(x+5)+2`

Correct

Great Work!

Incorrect

Horizontal translations (`\color(royalblue)(h)`) and vertical translations (`\color(forestgreen)(c)`) of a function here is in the form of `y=log(x-color(blue)(h))+color(forestgreen)(c)` where the point `( color(royalblue)(h), color(forestgreen)(c))` is on the function. Horizontal translations have the opposite sign as the direction of their shift.

`(-h) \ bb(rarr)` Shift Right

`(+h) \ bb(larr)` Shift Left

`(-c) \ bb(darr)` Shift Down

`(+c) \ bb(uarr)` Shift Up

To obtain the equation of the function by using `y=logx`, first shift the graph to the left by `5` units.

This shift is a horizontal translation left, where `color(blue)(h=+5)` inside the brackets.

Next, shift the graph down by `color(forestgreen)(-2)` units.

This shift is a vertical translation (`c`) where `color(forestgreen)(c=-2)`.

`y=`		`logx`	Write the new equation using `y=logx`, `color(forestgreen)(c=-2)`, and `color(blue)(h=+5)`. Remember horizontal translations have the opposite sign as the direction of their shift.
`y=`		`log(x+color(blue)(5))-color(forestgreen)(2)`

`y=log(x+5)-2`

Question 3 of 6

3. Question

Find the equation of the function when `x^2 + y^2 =9` is shifted (translated) to the right by `2` units and up by `3` units.

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

Correct

Great Work!

Incorrect

Horizontal translations (`\color(royalblue)(h)`) and vertical translations (`\color(forestgreen)(c)`) for a circle have the form `(x-color(blue)(h))^2 + (y-color(forestgreen)(c))^2=r^2` where the point `( color(royalblue)(h), color(forestgreen)(c))` is the center. Horizontal and vertical translations for circles have the opposite sign as the direction of their shift.

`(-h) \ bb(rarr)` Shift Right

`(+h) \ bb(larr)` Shift Left

`(-c) \ bb(uarr)` Shift Up

`(+c) \ bb(darr)` Shift Down

To obtain the equation of the function by using `x^2 + y^2=r^2`, first shift the graph to the right by `2` units. Remember circle equations have the form `(x-color(blue)(h))^2 + (y-color(forestgreen)(c))^2=r^2`.

This shift is a horizontal translation (`h`) (`2` units to the right) where `color(blue)(h=-2)` inside the brackets.

Next, shift the graph vertically up by `3` units where `color(forestgreen)(c=-3)` inside the brackets.

`x^2 + y^2=`		`9`	Since it’s in the form `(x-color(blue)(h))^2 + (y-color(forestgreen)(c))^2=r^2`, then `(x-color(blue)(2))^2 + (y-color(forestgreen)(3))^2=9`. Remember with circles, that the vertical and horizontal translations have the opposite sign since they are inside the brackets.
`(x-color(blue)(2))^2 + (y-color(forestgreen)(3))^2=`		`9`

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

Question 4 of 6

4. Question

Find the equation of the function when `y=sqrt(x)` is shifted (translated) to the left by `2` units and up by `4` units.

`y=sqrt(x+2)+4`
`y=sqrt(x-2)+4`
`y=sqrt(x-2)-4`
`y=sqrt(x+2)-4`

Correct

Great Work!

Incorrect

Horizontal translations (`\color(royalblue)(h)`) and vertical translations (`\color(forestgreen)(c)`) for a square root function has the form `y=sqrt(x-color(blue)(h))+color(forestgreen)(c)` where the point `( color(royalblue)(h), color(forestgreen)(c))` is the vertex for this function. Horizontal translations have the opposite sign as the direction of their shift.

`(-h) \ bb(rarr)` Shift Right

`(+h) \ bb(larr)` Shift Left

`(-c) \ bb(darr)` Shift Down

`(+c) \ bb(uarr)` Shift Up

To obtain the equation of the function by using `y=sqrt(x)`, first shift the graph to the left by `2` units. Remember square root functions have the form `y=sqrt(x-color(blue)(h))+color(forestgreen)(c)`.

This shift is a horizontal translation (`h`) where `color(blue)(h=+2)` inside the square root.

Next, shift the graph up by `4` units `color(forestgreen)((c=4))`

`y=`		`sqrt(x)`	Write the new equation using `y=sqrt(x)`, `color(forestgreen)(c=4)`, and `color(blue)(h=+2)`. Remember horizontal translations have the opposite sign as the direction of their shift.
`y=`		`sqrt(x+color(blue)(2))+color(forestgreen)(4)`

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

Question 5 of 6

5. Question

Find the equation of the function when `y=1/x` is shifted (translated) to the left by `3` units and up by `2` units.

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

Correct

Great Work!

Incorrect

Horizontal translations (`\color(royalblue)(h)`) and vertical translations (`\color(forestgreen)(c)`) for a hyperbolic function is in the form `y=1/(x-color(blue)(h))+color(forestgreen)(c)` where `( color(royalblue)(h), color(forestgreen)(c))` is the origin for this function. Horizontal translations have the opposite sign as the direction of their shift.

`(-h) \ bb(rarr)` Shift Right

`(+h) \ bb(larr)` Shift Left

`(-c) \ bb(darr)` Shift Down

`(+c) \ bb(uarr)` Shift Up

To obtain the equation of the function by using `y=1/x`, first shift the graph to the left by `3` units. Remember hyperbolas have the form `y=1/(x-color(blue)(h))+color(forestgreen)(c)`.

Firstly, we shift a horizontal translation (left), where `color(blue)(h=+3)` in the denominator.

Next, shift a vertical translation up by `2` units `color(forestgreen)((c=2))`.

`y=`		`1/x`	Write the new equation using `y=1/x`, `color(forestgreen)(c=2)`, and `color(blue)(h=+3)`. Remember horizontal translations have the opposite sign as the direction of their shift and hyperbolas have the form `y=1/(x-color(blue)(h))+color(forestgreen)(c)`.
`y=`		`1/(x+color(blue)(3))+color(forestgreen)(2)`

`y=1/(x+3)+2`

Question 6 of 6

6. Question

Find the equation of the function when `y=2^(x-2)` is shifted (translated) to the left by `3` units and up by `4` units.

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

Correct

Great Work!

Incorrect

Horizontal translations (`\color(royalblue)(h)`) and vertical translations (`\color(forestgreen)(c)`) for an exponential function have the form `y=2^(x-color(blue)(h)) + color(forestgreen)(c)` where the point `( color(royalblue)(h), color(forestgreen)(c))` is on the function. Horizontal translations have the opposite sign as the direction of their shift.

`(-h) \ bb(rarr)` Shift Right

`(+h) \ bb(larr)` Shift Left

`(-c) \ bb(darr)` Shift Down

`(+c) \ bb(uarr)` Shift Up

To obtain the equation of the function by using `y=2^(x-2)`, first shift the graph to the left by `3` units. Remember exponential functions have the form `y=2^(x-color(blue)(h)) + color(forestgreen)(c)`.

This shift is a horizontal translation (left), where `color(blue)(h=+3)`.

Next, is a vertical translation up by `4` units `color(forestgreen)((c=4))`.

`y=`	`2^(x-2)`	Write the new equation using `y=2^(x-2)`, `color(forestgreen)(c=4)`, and `color(blue)(h=+3)`. Remember horizontal translations have the opposite sign as the direction of their shift and exponential functions have the form `y=2^(x-color(blue)(h)) + color(forestgreen)(c)`.
`y=`	`2^(x-2+color(blue)(3))+color(forestgreen)(4)`	Simplifying the exponent.
`y=`	`2^(x+1)+4`

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

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