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Combinations of Transformations: OrderCombinations of Transformations: Order
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Question 1 of 8
1. Question
When transforming `y=x^3` to `y=3x^3 -2` is the vertical dilation factor of `3` or the vertical translation of `2` unit down applied first?
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Order is important when a vertical dilation is combined with a vertical translation.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.A standard function form for a vertical dilation factor is `y=color(blue)(k)f(x)` where `color(blue)(k)` is the vertical dilation factor.In order to find the order of the transformations we will try. First, lets try a vertical dilation in order to transform `y = x^3` to `y= 3x^3 -2`.The first transformation to be applied on `y=x^3 ` will be the vertical dilation factor `color(blue)(k)` where `color(blue)(k=3)`.`y=` `color(blue)(3)x^3` Apply the vertical dilation factor `color(blue)(k=3)`. Remember `y=color(blue)(k)f(x)`. The second transformation to be applied on `y=3x^3 ` will be the vertical translation `y=f(x)+color(purple)(c)` where `color(purple)(c= – 2)`.`y=` `3x^3 color(purple)(- 2)` Apply the vertical translation `color(purple)(c= – 2)`. Remember `y=f(x)+color(purple)(c)`. The vertical dilation factor is applied first. -
Question 2 of 8
2. Question
When transforming `y=x^3` to `y=(5x-1)^3`, which comes first. The horizontal dilation factor of `1/5` or the horizontal translation of `1` unit to the right?
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Order is important when a horizontal dilation is combined with a horizontal translation.A standard function form for a horizontal translation is `y=f(x+color(red)(h))` where `color(red)(+h)` is a left movement along the `x`-axis.The application of the horizontal dilation `color(blue)(text{factor})` on a `x` variable is `color(blue)(text{factor})=1/color(green)(a)` and we can apply `x/color(blue)(text{factor})`.To find the correct order we will try one of two ways.To go from `y=x^3` to `y=(5x-1)^3`, lets try the horizontal scale factor first.First let’s apply the Horizontal Scale `color(blue)(\text(Factor)=1/5)`.`y` `=` `x^3` Then we use `x/\text(factor)` where we divide `x` by `color(blue)(1/5)` `y` `=` `(x/(color(blue)(1/5)))^3` Simplify. `y` `=` `(5x)^3` Secondly, we apply the horizontal translation of `1` unit to the rightWe will use `y=f(x color(red)(+h))` where `+color(red)(+h)` is moving to the leftNotice `color(red)(h=-1)` when we move to the rightWe replace `x` for `(x color(red)(-1))` in `y=(5x)^3`.It becomes `y=(5(x color(red)(-1)))^3`This clearly is incorrect as it does not match `y=(5x-1)^3`. Which means the order is incorrect.Now lets try another order.Starting with `y=x^3`.First we will apply the horizontal translation `h` to `y=x^3` where `color(red)(h=-1)` (moving to the right by `1` unit). Now we replace `x` for `(x color(red)(-1))` in `y=x^3`It simplifies to `y=(x color(red)(-1))^3`.The second transformation to be applied will be the horizontal scale factor where `color(blue)(\text(factor)=1/5)`.We will apply `x/\text(factor)` which becomes `y=(x/(color(blue)(1/5))color(red)(-1))^3` this simplifies to:`y=(5x color(red)(-1))^3`Yes, this matches the original function!Correct order: first horizontal translation then horizontal dilationThe horizontal translation is applied first. -
Question 3 of 8
3. Question
When transforming `y=lnx` to `y=ln[4(x+5)]` which comes first. The horizontal dilation factor of `1/4` or the horizontal translation of `5` units to the left?
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Order is important when a horizontal dilation is combined with a horizontal translation.A standard function form for a horizontal translation is `y=f(x+color(red)(h))` where `color(red)(+h)` is a left movement along the `x`-axis.The application of the horizontal dilation `color(blue)(text{factor})` on a `x` variable is `color(blue)(text{factor})=1/color(green)(a)` and we can apply `x/color(blue)(text{factor})`.Lets try the order that the transformations will occur so `y=ln x` can become `y=ln[4(x+5)]`.The first transformation to be applied to `y=lnx` will be the horizontal dilation `color(blue)(text{factor}=1/4)`.`y=` `ln x` We will apply `x/color(blue)(text{factor})`, take the `x` and divide by `1/4`. `y=` `ln (x/(1/4))` Simplify `y=` `ln 4x` The second transformation to be applied to `y=ln 4x` will be the horizontal translation `color(red)(h)` where `color(red)(h=+5)` since it is moving to the left by `5` units.`y=` `ln[4(x + color(red)(5))]` Apply the horizontal translation `color(red)(h=+5)`. Remember `y=f(x+color(red)(h))`. Yes, this is the correct order! It matches the original function.The horizontal dilation is applied first. -
Question 4 of 8
4. Question
When transforming `y=x^2` to `y=(1/5x + 3)^2` is the horizontal dilation factor of `5` or the horizontal translation of `3` units left applied first?
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Order is important when a horizontal dilation is combined with a horizontal translation.A standard function form for a horizontal translation is `y=f(x+color(red)(h))` where `color(red)(+h)` is a left movement along the `x`-axis.The application of the horizontal dilation `color(blue)(text{factor})` on a `x` variable is `color(blue)(text{factor})=1/color(green)(a)` and we can apply `x/color(blue)(text{factor})`.To find the correct order where `y=x^2` becomes `y=(1/5x + 3)^2` we are going to try a horizontal translation.The first transformation to be applied to `y=x^2` will be the horizontal translation `color(red)(h)` where `color(red)(h=+3)`.`y=` `(x+color(red)(3))^2` Apply the horizontal translation `color(red)(h=+3).` Remember `y=f(x+color(red)(h))` where `color(red)(+h)` is moving to the left (opposite). The second transformation to be applied to `y=(x+3)^2` will be the horizontal dilation `color(blue)(text{factor}=5)`.`y=` `(x/color(blue)(5))+3)^2` Apply the horizontal dilation `color(blue)(\text(factor)=5)`. We apply `x/color(blue)(text{factor})` by taking the `x` and dividing it by `5`. `y=` `(1/5x+3)^2` Yes, this is the correct order! It matches.The horizontal translation is applied first. -
Question 5 of 8
5. Question
Describe the order of the transformations when transforming `y=e^x` to `y=3e^(6x) -1`.
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Order is important when a vertical dilation is combined with a vertical translation.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 `color(blue)(text{factor})=1/color(green)(a)` and we can apply `x/color(blue)(text{factor})`.A standard function form for a vertical dilation factor is `y=color(red)(k)f(x)` where `color(red)(k)` is the vertical dilation factor.To find the correct order so we can go from `y=e^x` to `y=3e^(6x) -1`, lets try a horizontal dilation.The first transformation to be applied to `y=e^x` will be the horizontal dilation `color(blue)(text{factor})=1/color(green)(a)` where `color(green)(a=6)`.
Notice `color(blue)(\text(factor)=1/6)`.This is from the given equation `y=3e^(color(green)(6)x)-1`.This is in the form `y=e^(color(green)(a)x)` where `a=6`, so:`y=` `e^(color(green)(6)x)` Alternatively, we could use `x/color(blue)(text{factor})` which equals `e^(x/(color(blue)(1/6)))` which equals: `y=e^(color(green)(6)x)`.The second transformation to be applied on `y=e^(6x) ` will be the vertical dilation factor `color(red)(k)` where `color(red)(k=3)`.`y=` `color(red)(3)e^(6x)` Apply the vertical dilation factor `color(red)(k=3)`. Remember `y=color(red)(k)f(x)`. The third transformation to be applied on `y=3e^(6x) ` will be the vertical translation `y=f(x)+color(purple)(c)` where `color(purple)(c=-1)`.`y=` `3e^(6x)color(purple)(-1)` Apply the vertical translation `color(purple)(c=-1)`. Yes, this is the correct order! It matches the original function.1. Horizontal dilation, 2. Vertical dilation, 3. Vertical translation -
Question 6 of 8
6. Question
Describe the order of the transformations when transforming `y=logx` to `y=7log(x+5)-4`.
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Great Work!
Incorrect
Order is important when a vertical dilation is combined with a vertical translation.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.A standard function form for a horizontal translation is `y=f(x+color(blue)(h))` where `color(blue)(h)` is a shift to the left along the `x`-axis.A standard function form for a vertical dilation factor is `y=color(red)(k)f(x)` where `color(red)(k)` is the vertical dilation factor.To find the correct order we will try.To go from `y=logx` to `y=7log(x+5)-4` lets try a horizontal translation.The first transformation to be applied to `y=logx` will be the horizontal translation `color(blue)(h)` where `color(blue)(h=+5)`.`y=` `log(xcolor(blue)(+5))` Apply the horizontal translation `color(blue)(h=+5)`. Remember `y=f(x+color(blue)(h))`. The second transformation to be applied on `y=log(x+5) ` will be the vertical dilation factor `color(red)(k)` where `color(red)(k=7)`.`y=` `color(red)(7)log(x+5)` Apply the vertical dilation factor `color(red)(k=7)`. Remember `y=color(red)(k)f(x)`. The third transformation to be applied on `y=7log(x+5) ` will be the vertical translation `y=f(x)+color(purple)(c)` where `color(purple)(c=-4)`.`y=` `7log(x+5)color(purple)(-4)` Apply the vertical translation `color(purple)(c=-4)`. Remember `y=f(x)+color(purple)(c)`. Yes, this is the correct order! It matches the original function.1. Horizontal translation, 2. Vertical dilation, 3. Vertical translation -
Question 7 of 8
7. Question
Describe the order of the transformations when transforming `y=|x|` to `y=|-4(x+2)|+6`.
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Order is important when a horizontal dilation is combined with a horizontal translation.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.A standard function form for a horizontal translation is `y=f(x+color(red)(h))` where `color(red)(+h)` is a left movement along the `x`-axis.The application of the horizontal dilation (`color(blue)(text{factor})`) on a `x` variable is `color(blue)(text{factor})=1/color(green)(a)` or we can use `x/color(blue)(text{factor})`.A standard function form for a reflection about the y-axis is `y=f(-x)` where we replace `x` for `-x`.To find the correct order we will take a guess.To go from `y=|x|` to `y=|-4(x+2)|+6` let’s try a reflection about the y-axis.The first transformation to be applied to `y=|x|` will be the reflection about the y-axis.`y=` `|-x|` Simply replace `x` for `-x` The second transformation to be applied to `y=|-x|` will be the horizontal dilation.We can use `color(blue)(text{factor})=1/color(green)(a)`.This is in the form `y=|ax|`. Where `a=4`Therefore `y=|-4x|`Alternatively, we can use `color(blue)(\text(factor)=1/a)` which equals `1/4`.
Therefore horizontal `color(blue)(\text(factor)=1/4)`Now using `x/(color(blue)(\text(factor)))=|(x)/(color(blue)(1/4))|`Therefore `y=` `|-color(green)(4)x|` The third transformation to be applied to `y=|-4x|` will be the horizontal translation `color(red)(h)` where `color(red)(h=+2)` (where it is `2` units to the left).`y=` `|-4(xcolor(red)(+2))|` Apply the horizontal translation `color(red)(h=+2)`. Remember `y=f(x+color(red)(h))`. The fourth transformation to be applied on `y=|-4(x+2)| ` will be the vertical translation `y=f(x)+color(purple)(c)` where `color(purple)(c=+6)`.`y=` `|-4(x+2)|+color(purple)(6)` Yes, this is the correct order! It matches the original function.1. Reflection about the `y`-axis 2. Horizontal dilation, 3. Horizontal translation, 4. Vertical translation -
Question 8 of 8
8. Question
Describe the order of the transformations when transforming `y=x^3` to `y=8(3x-9)^3 +2`.
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Order is important is when horizontal dilation is combined with a horizontal translation. Order is also important when a vertical dilation is combined with a vertical translationA 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.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.A standard function form for a vertical dilation factor is `y=color(orange)(k)f(x)` where `color(orange)(k)` is the vertical dilation factor.The application of the horizontal dilation `color(blue)(text{factor})` on a `x` variable is `color(blue)(text{factor})=1/color(green)(a)` or we can apply `x/color(blue)(text{factor})`.The universal formula form with two translations and two dilations factors in one equation is `y = color(orange)(k)f(color(green)(a)(x+color(red)(b)))+color(purple)(c)`, where `color(red)(b)` is also `color(red)(h)` in the horizontal translation.
`color(orange)(k)` is the vertical dilation factor. `color(red)(+b)` is shift left. `color(purple)(+c)` is shift up. Horizontal factor is `1/color(green)(a).`To find the correct order we will take a guess to go from `y=x^3` to `y=8(3x-9)^3 + 2`. First we are going to factor it out.`y` `=` `8(3(x-3))^3+2` Factor out `3` from `3x-9`. Now it is in the universal formula form `y= color(orange)(k)f(color(green)(a)(x+color(red)(b)))+color(purple)(c)`. The first transformation to be applied to `y=x^3` will be the horizontal dilation `color(blue)(text{factor})=1/color(green)(a)` where `color(green)(a=3)` (from universal formula).`y=` `(color(green)(3)x)^3` Apply the horizontal parameter `color(green)(a=3)`. Remember `color(blue)(text{factor})=1/color(green)(a)` and `x/color(blue)(text{factor})`. The second transformation to be applied to `y=(3x)^3` will be the horizontal translation `color(red)(h)` where `color(red)(h=-3)`. Simply replace `x` for `x-3`.`y=` `(3(xcolor(red)( – 3)))^3` Apply the horizontal translation `color(red)(h=-3)`. Remember `y=f(x+color(red)(h))`. The third transformation to be applied on `y=(3(x-3))^3 ` will be the vertical dilation factor `color(orange)(k)` where `color(orange)(k=8)`.`y=` `color(orange)(8)(3(x-3))^3` Apply the vertical dilation factor `color(orange)(k=8)`. Remember `y=color(orange)(k)f(x)`. The fourth transformation to be applied on `y=8(3(x-3))^3 ` will be the vertical translation `y=f(x)+color(purple)(c)` where `color(purple)(c=+2)`.`y=` `8(3(x-3))^3+color(purple)(2)` Apply the vertical translation `color(purple)(c=2)`. Remember `y=f(x)+color(purple)(c)`. Yes, this is the correct order! It matches the original function.1. Horizontal dilation, 2. Horizontal translation, 3. Vertical dilation, 4. Vertical translation
Quizzes
- Vertical Translations (Shifts) 1
- Vertical Translations (Shifts) 2
- Vertical Translations (Shifts) from a Point
- Horizontal Translations (Shifts) 1
- Horizontal Translations (Shifts) from a Point
- Horizontal Translations (Shifts) from a Graph
- Horizontal and Verticals Translations (Shifts) from a Graph
- Sketch a Graph using Translations (Shifts)
- Write the Equation from a Graph
- Write the Equation from Translations (Shifts) 1
- Vertical Dilations (Stretch/Shrink)
- Horizontal Dilations (Stretch/Shrink) 1
- Horizontal Dilations (Stretch/Shrink) 2
- Horizontal Dilations (Stretch/Shrink) – Scale Factor
- Horizontal and Vertical Dilations (Stretch/Shrink) 1
- Horizontal and Vertical Dilations (Stretch/Shrink) 2
- Horizontal and Vertical Dilations (Stretch/Shrink) 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