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Transformations of Functions>
Combinations of Transformations: Find Equation>
Combinations of Transformations: Find Equation 2Combinations of Transformations: Find Equation 2
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Question 1 of 6
1. Question
Find the transformed version of `y=1/x` when a horizontal translation of `3` units left, a vertical dilation of `2`, and a vertical translation of `4` units down are applied.
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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.A standard function form for a vertical dilation is `color(blue)(k)f(x)` where `color(blue)(k)` is the vertical dilation factor.To transform `y=1/x` with a horizontal translation of `color(red)(3)` units left, a vertical dilation of `color(blue)(2)`, and a vertical translation of `color(purple)(4)` units down, first apply a horizontal translation moving to the left (negative) which means that `color(red)(+h)` is a positive. Do this by using `y=f(x+color(red)(h))` and `color(red)(h=3)`.`y=` `1/x` Apply the horizontal translation of `color(red)(h=3)`. Remember `y=f(x+color(red)(h))`. `=` `1/(xcolor(red)(+3))` Simplify `=` `1/(x+3)` Now apply the vertical dilation of `color(blue)(k=2)`. Use `color(blue)(k)f(x)`.`y=` `1/(x+3)` Apply the vertical dilation of `color(blue)(k=2)`. Use `color(blue)(k)f(x)`. `=` `color(blue)(2)times(1/(x+3))` Simplify `=` `2/(x+3)` Finally apply the vertical translation of `color(purple)(4)` units down. Use `y=f(x)+color(purple)(c)` where `color(purple)(c)` is an upward movement along the `y`axis. In this case `color(purple)(c=4)` because we are going downward.`y=` `2/(x+3)` Apply the vertical translation of `color(purple)(4)` units down. This means `color(purple)(c=4)`. `=` `2/(x+3)color(purple)(4)` Simplify `=` `2/(x+3)4` `y=2/(x+3)4` 
Question 2 of 6
2. Question
Find the equation when `y=x^3` is transformed with a horizontal dilation factor of `2` then a vertical translation of `3` units up.
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The application of the horizontal dilation (`color(blue)(text{factor})`) on a `x` variable is `x/color(blue)(text{factor})`.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.To determine the transformed equation, start by applying the horizontal dilation factor of `color(blue)(2)` first. Do this by using `x/color(blue)(text{factor})` and `color(blue)(text{factor})color(blue)(=2)`.`y=` `x^3` Apply the horizontal dilation factor of `color(blue)(2)`. Remember `x/color(blue)(text{factor})`. `=` `(x/{color(blue)(2)})^3 ` Simplify `=` `(x/2)^3` Apply the vertical translation of `color(purple)(3)` units up. Use `y=f(x)+color(purple)(c)` where `color(purple)(c)` is an upward movement along the `y`axis. This means `color(purple)(c=3)`. `=` `(x/2)^3 color(purple)(+3)` Simplify `=` `(x/2)^3 +3` Simplify. `y=` `x^3/8 +3` Simplify. `y=x^3/8 +3` 
Question 3 of 6
3. Question
Find the equation when `y=sqrt(x)` is reflected about the `y`axis then transformed with a vertical dilation factor of `2`, then a horizontal dilation factor of `1/3`.
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Reflections about the `y`axis have the property where we replace `x\rightarrowx`.
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 (`color(green)(text{factor})`) on a `x` variable is `x/color(green)(text{factor})`.To determine the transformed equation, start by applying the reflection around the `y`axis by replacing `x\rightarrowx`.`y=` `sqrt(x)` Replace `x` by `x`. `=` `color(blue)(2)sqrt(x)` Apply the vertical dilation of `color(blue)(k=2)`. Use `color(blue)(k)f(x)`. `=` `2sqrt(x)` Apply the horizontal dilation factor of `color(green)(1/3)`. Remember `x/color(green)(text{factor})` `=` `2sqrt(x/color(green)(1/3))` Simplify `y=` `2sqrt(3x)` `y=2sqrt(3x)` 
Question 4 of 6
4. Question
Find the transformed equation when the original function `y=logx` is translated horizontally by `2` units to the right, then transformed with the horizontal dilation factor of `3`.
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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.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)(2)` units right. Use `y=f(x+color(red)(h))` where `color(red)(h)` is a shift to the left along the `x`axis. This means `color(red)(h=2)`.`y=` `logx` Apply the horizontal translation of `color(red)(2)` units right. Use `y=f(xcolor(red)(2))`. `=` `log(xcolor(red)(2))` Simplify `=` `log(x2)` Apply the horizontal dilation factor of `color(blue)(3)`. Remember `x/color(blue)(text{factor})`. `=` `log((x/color(blue)(3)) 2)` Simplify `y=` `log(1/3x – 2)` `y=log(1/3x – 2)` 
Question 5 of 6
5. Question
Find the transformed version of `y=3^x` when a vertical translation of `2` units up, a horizontal translation of `2` units to the right, and a vertical dilation factor of `1` are applied.
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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.A standard function form for a vertical dilation is `color(blue)(k)f(x)` where `color(blue)(k)` is the vertical dilation.To transform `y=3^x` with a vertical translation of `color(purple)(2)` units up, horizontal translation of `color(red)(2)` units right, and a vertical dilation of `color(blue)(1)`, first apply a vertical translation of `2` units up. This means `color(purple)(c=2)`.`y=` `3^x` Apply the vertical translation of `color(purple)(2)` units up. 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)`. `=` `3^x + color(purple)(2)` Simplify `=` `3^x + 2` Next, remember `color(red)(+h)` (left) is a shift to the left along the `x`axis. So moving to the right, `color(red)(h)` is a negative. Do this by using `y=f(x+color(red)(h))` and `color(red)(h=2)`.`y=` `3^x + 2` Apply the horizontal translation of `color(red)(h=2)`. Remember `y=f(x+color(red)(h))`. `=` `3^(xcolor(red)(2)) + 2` Simplify `=` `3^(x2) + 2` Now apply the vertical dilation of `color(blue)(k=1)`. Use `color(blue)(k)f(x)`.`y=` `3^(x2) + 2` Apply the vertical dilation of `color(blue)(k=1)`. Use `color(blue)(k)f(x)`. `=` `color(blue)(1)[3^(x2) + 2]` Simplify `=` `3^(x2) – 2` `y=3^(x2) – 2` 
Question 6 of 6
6. Question
Find the equation when `y=sqrt(x)` is transformed with a vertical translation of `3` units up, a horizontal dilation factor of `2`, a horizontal translation of `1` unit up, and a reflection about the `x`axis.
Correct
Great Work!
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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})`.Reflections about the `x`axis have the property where we replace `y\rightarrowy`.To determine the transformed equation, start by applying the vertical translation of `color(purple)(3)` units up. Use `y=f(x)+color(purple)(c)` where `color(purple)(+c)` is an upward movement along the `y`axis. This means `color(purple)(c=3)` .`y=` `sqrt(x)color(purple)(+3)` Apply the vertical translation `color(purple)(c=3)`. `=` `sqrt(x/color(blue)(2))+3` Apply the horizontal dilation factor of `color(blue)(2)`. Remember `x/color(blue)(text{factor})`. `=` `sqrt(1/2(xcolor(red)(1)))+3` Apply the horizontal translation `color(red)(1)` unit right. `=` `[sqrt(1/2(x1))+3]` Reflecting about the `x`axis. Replace `y` for `y`. `y=` `sqrt(1/2(x1))3` `y=sqrt(1/2(x1))3`
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