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Combinations of Transformations: Find Equation>
Combinations of Transformations: Find Equation 3Combinations of Transformations: Find Equation 3
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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.
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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.
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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`.
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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.
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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})`.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`.
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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`.
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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 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 (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