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Combinations of Transformations: Find Equation>
Combinations of Transformations: Find Equation 1Combinations of Transformations: Find Equation 1
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Question 1 of 5
1. Question
Find the equation when `y=x^2` is shifted `3` units down and `4` units to the right.
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A standard function form for translations and dilations is `y=color(red)(a)(x-color(blue)(h))^2+color(purple)(c)` where:- `color(red)(a)` is the vertical dilation (sometimes we use `y=color(red)(k)(x-color(blue)(h))^2+color(purple)(c)`)
- `color(blue)(h)` is the `x`-coordinate of the vertex
- `color(purple)(c)` is the `y`-coordinate of the vertex
- Vertex `(color(blue)(h),color(purple)(c))`
To find the equation when `y=x^2` is shifted `3` units down and `4` units right, remember `y=color(red)(a)(x-color(blue)(h))^2+color(purple)(c)` where `color(red)(a)` is the vertical dilation, `color(blue)(h)` is the `x`-coordinate of the vertex, and `color(purple)(c)` is the `y`-coordinate of the vertex.The vertex in `y=x^2` (which can also be written as `y=(x-0)^2+0)` is `(0,0)`. If you shift the vertex `color(purple)(3)` units down (downward movement along the `y`-axis) and `color(blue)(4)` units right (shift to the left along the `x`-axis) you will get a new vertex of `(0+color(blue)(4),0+(color(purple)(-3)))=(color(blue)(4),color(purple)(-3))`To get the new equation, substitute the new vertex coordinates `(color(blue)(4),color(purple)(-3))` into `y=(x-0)^2+0`.
This gives `y=(x-color(blue)(4))^2-color(purple)(3)` which simplifies to `y=(x-4)^2-3`.`y=(x-4)^2-3` -
Question 2 of 5
2. Question
Find the equation when `(x-1)^2 + (y-3)^2 = 1` is reflected about the `y` axis and then shifted down `2` units.
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Reflections about the `y`-axis means that we have to replace `x\rightarrow-x`.A standard circle equation is in the form: `(x-color(blue)(h))^2 + (y-color(purple)(c))^2 =r^2`- `color(blue)(h)` is the `x`-coordinate for the centre
- `color(purple)(c)` is the `y`-coordinate for the centre
First we find the equation when `(x-1)^2 + (y-3)^2 = 1` is reflected about the `y`-axis, we replace the `x` by `-x`. We get `(-x-1)^2 + (y-3)^2 = 1` or simply `(x+1)^2 + (y-3)^2 = 1`.Remember `(x-color(blue)(h))^2 + (y-color(purple)(c))^2 =r^2` where `color(blue)(h)` is the `x`-coordinate of the centre, and `color(purple)(c)` is the `y`-coordinate of the centre.The center for `(x+1)^2 + (y-3)^2 = 1` is `(-1,3)`.Next we will shift this circle `2` units down along the `y-`axis. Since the formula indicates `(y-c)^2` it becomes `(y- -c)^2 \rightarrow (y+c)^2.`Notice it is `color(purple)(+c)` even though we are going downwards. So as a rule of thumb for this formula: `(x-color(blue)(h))^2 + (y-color(purple)(c))^2 =r^2` anytime `color(purple)(-c)` or `color(blue)(-h)` is inside the brackets we move in the opposite direction.This gives `(x+1)^2+(y-1)^2=1.`
The centre is `(-1,1)`.`(x+1)^2 + (y-1)^2 = 1` -
Question 3 of 5
3. Question
Find the equation when `y=e^x` is vertically dilated by a factor of `4` and shifted `5` units to the left.
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A standard exponential function form for translations and dilations is- `color(red)(k)` is the vertical dilation `y=color(red)(k)e^((x+color(blue)(h)))`
- `color(blue)(h)` is the `x`-coordinate for the horizontal translation
To find the equation when `y=e^x` is vertically dilated by a factor of `4` and shifted `5` units left, remember `y=color(red)(k)e^((x+color(blue)(h)))` where `color(red)(k)` is the vertical dilation and `color(blue)(h)` is the horizontal translation.The equation `y=e^x` when vertically dilated by a factor of `4` becomes `y=4e^x`. Then applying the translation of `5` units to the left
(left means `+5`), `y=4e^x` becomes `y=4e^(x+5)`.`y=4e^(x+5)` -
Question 4 of 5
4. Question
Find the equation when `y=2x^2 -3x` is shifted `1` unit left and then reflected about the `y` axis.
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Reflections about the `y`-axis means that we have to replace `x\rightarrow color(red)(-x)`.A standard function for a horizontal translation is `y=(x+color(blue)(h))` where `color(blue)(+h)` is a left shift along the x-axis.To find the equation when `y=2x^2 -3x` is shifted `1` unit left. Now, in order to shift it `1` unit to the left we replace `x` with `(x color(blue)(+1))`.`y` `=` `2(x color(blue)(+1))^2-3(x color(blue)(+1))` `y` `=` `2(x^2+2x+1)-3x-3` `y` `=` `2x^2+4x+2-3x-3` `y` `=` `2x^2+x-1` Then we reflect the equation `y=2x^2+x-1` about the `y`-axis by replacing `x` with `color(red)(-x)`. The equation becomes, `y=2(color(red)(-x))^2+(color(red)(-x))-1` or simply `y=2x^2-x-1`.`y=2x^2-x-1` -
Question 5 of 5
5. Question
Find the transformed version of `y=x^3` when you have a horizontal dilation factor of `1/2`, a horizontal translation of `3` units left, and a vertical dilation of `4` are applied.
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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 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 dilation is `color(blue)(k)f(x)` where `color(blue)(k)` is the vertical dilation.To transform `y=x^3` with a horizontal dilation factor of `color(blue)(1/2)`, horizontal translation of `color(red)(3)` units left, and a vertical dilation of `color(blue)(4)`, start by applying the horizontal dilation factor of `color(blue)(1/2)` first. Do this by using `x/color(blue)(text{factor})` and `color(blue)(text{factor})color(blue)(=1/2)`.`y=` `(x/color(blue)(1/2))^3` Apply the horizontal dilation factor of `color(blue)(1/2)`. Remember `x/color(blue)(text{factor})`. `=` `(2x)^3` Simplify `=` `(2(xcolor(red)(+3)))^3` Apply the horizontal translation of `color(red)(3)` units left. Use `y=f(xcolor(red)(+3))`. `=` `(2(x+3))^3` Simplify Now apply the vertical dilation of `color(blue)(k=4)`. Use `color(blue)(k)f(x)`.`y=` `color(blue)(4)(2(x+3))^3` Apply the vertical dilation of `color(blue)(k=4)`. Use `color(blue)(k)f(x)`. `=` `4(2(x+3))^3` Simplify `=` `32(x+3)^3` `y=32(x+3)^3`
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