Horizontal Translations from a Graph

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Question 1 of 5

1. Question

Find the equation of the graph on the right of `f(x)=x^3 – 2x^2 +x`

Correct

Great Work!

Incorrect

Horizontal translations (shifts) of functions are written in the form `y=f(x+ color(royalblue)(h))`.

`color(royalblue)(h)` is how many units right or left the graph will be shifted.

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

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

To find the equation of the graph on the right of `f(x)=x^3 – 2x^2 +x`, count the number of units the graph is shifted over.

Count the number of units the graph is shifted over to the right.

The graph is shifted by `2` units.

Using the form `y=f(x+ color(royalblue)(h))` for horizontal translations and remembering that a shift to the right means `color(royalblue)(h)` is negative, the formula for the new graph to the right is `f(x)=(x color(royalblue)(-2))^3 – 2(x color(royalblue)(-2))^2 +(x color(royalblue)(-2))`

Then, we simplify the function by using the distributive property.

`f(x)`	`=`	`(x-2)^3 – 2(x-2)^2 +(x-2)`
	`=`	`(x-2)(x-2)(x-2)- 2(x-2)(x-2) +(x-2)`
	`=`	`(x-2)(x^2 -4x +4)- 2(x^2 -4x +4) +(x-2)`
	`=`	`x^3 – 4x^2 +4x – 2x^2 +8x – 8 -2x^2 + 8x – 8 + x – 2`
	`=`	`x^3 – 8x^2 + 21x -18`	Combining similar terms

Therefore, our simplified formula for the new graph to the right is `f(x)=x^3 – 8x^2 + 21x -18`

Question 2 of 5

2. Question
Find the equation of the graph on the left of `f(x)=x^2`
Correct

Great Work!

Incorrect

Horizontal translations (shifts) of functions are written in the form `y=f(x+ color(royalblue)(h))`.

`color(royalblue)(h)` is how many units right or left the graph will be shifted.

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

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

To find the equation of the graph on the left of `f(x)=x^2`, count the number of units the graph is shifted over.

Count the number of units the graph is shifted over to the left.

The graph is shifted by `3` units.

Using the form `y=f(x+ color(royalblue)(h))` for horizontal translations and remembering that a shift to the left means `color(royalblue)(h)` is positive, the formula for the new graph to the left is `f(x)=(x+ color(royalblue)(3))^2`
Question 3 of 5

3. Question
Find the equation of the graph on the left of `f(x)=\sqrt{x}`
Correct

Great Work!

Incorrect

Horizontal translations (shifts) of functions are written in the form `y=f(x+ color(royalblue)(h))`.

`color(royalblue)(h)` is how many units right or left the graph will be shifted.

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

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

To find the equation of the graph on the left of `f(x)=\sqrt{x}`, count the number of units the graph is shifted over.

Count the number of units the graph is shifted over to the left.

The graph is shifted by `4` units.

Using the form `y=f(x+ color(royalblue)(h))` for horizontal translations and remembering that a shift to the left means `color(royalblue)(h)` is positive, the formula for the new graph to the left is `f(x)=\sqrt{x+4}`
Question 4 of 5

4. Question
Find the equation of the graph on the right of `f(x)=2^x`
Correct

Great Work!

Incorrect

Horizontal translations (shifts) of functions are written in the form `y=f(x+ color(royalblue)(h))`.

`color(royalblue)(h)` is how many units right or left the graph will be shifted.

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

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

To find the equation of the graph on the left of `f(x)=2^x`, count the number of units the graph is shifted over.

Count the number of units the graph is shifted over to the right.

The graph is shifted by `3` units.

Using the form `y=f(x+ color(royalblue)(h))` for horizontal translations and remembering that a shift to the right means `color(royalblue)(h)` is negative, the formula for the new graph to the right is `f(x)=2^(x color(royalblue)(-3))`
Question 5 of 5

5. Question
Find the equation of the graph on the right of `f(x)=|x+2|`
Correct

Great Work!

Incorrect

Horizontal translations (shifts) of functions are written in the form `y=f(x+ color(royalblue)(h))`.

`color(royalblue)(h)` is how many units right or left the graph will be shifted.

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

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

To find the equation of the graph on the left of `f(x)=|x+2|`, count the number of units the graph is shifted over.

Count the number of units the graph is shifted over to the right.

The graph is shifted by `5` units.

Using the form `y=f(x+ color(royalblue)(h))` for horizontal translations and remembering that a shift to the right means `color(royalblue)(h)` is negative, the formula for the new graph to the left is `f(x)=|x+ 2 color(royalblue)(-5)|` or simply `f(x)=|x – 3|`