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Transformations of Functions>
Reflection with Rotation>
Reflection with RotationReflection with Rotation
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Question 1 of 5
1. Question
Given `y=(x-2)^2 -1`.
Sketch `y=-f(-x)`
Correct
Great Work!
Incorrect
A rotation of `180` degrees (about the origin) is found when `y=f(x)` is transformed to `y=-f(-x)`.To be able to sketch the new function after the rotation of `180` degrees, find `y=-f(-x)`.`f(x)=` `(x-2)^2-1` `-f(-x)=` `color(red)(-)[((color(blue)(-x)-2)^2 – 1]` Transform `y=f(x)` into `y=-f(-x)`. Simplify inside the square brackets first. `=` `color(red)(-)[(x+2)^2 – 1]` `=` `-(x+2)^2 + 1` Set `-(x+2)^2 + 1` equal to zero and solve for `x`. This will give you the rotated `x`-intercepts.`(x+2)^2=` ` + 1` Adding `(x+2)^2` to both sides. `x+2=` `\pm 1` Taking the square root of both sides. `x=` `\pm 1 – 2` Subtracting `2`to both sides `x=` `1 – 2` For the positive `1`. `x=` `-1` This is the first point. `x=` `-1 – 2` For the negative `1`. `x=` `-3` This is the second point. Now plot these `x`-intercept points on the graph.
Rotate `180°` and sketch the original graph around the point `(0,0)` and going through `x`-intercept `x=-1`, and `x`-intercept `x=-3`.
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Question 2 of 5
2. Question
Given `y=(x+1)^2`.
Sketch `y=-f(-x)`
Correct
Great Work!
Incorrect
A rotation of `180` degrees (about the origin) is found when `y=f(x)` is transformed to `y=-f(-x)`.To be able to sketch the new function after the rotation of `180` degrees, find `y=-f(-x)`.`f(x)=` `(x+1)^2` `-f(-x)=` `color(red)(-)[((color(blue)(-x)+1)^2]` Transform `y=f(x)` into `y=-f(-x)`. Simplify inside the square brackets first. `=` `color(red)(-)[(x-1)^2]` `=` `-(x-1)^2` Set `-(x-1)^2` equal to zero and solve for `x`. This will give you the rotated `x`-intercepts.`(x-1)^2=` `0` Multiplying both sides by `-1`. `x-1=` `0` Taking the square root of both sides. `x=` `1` Adding `1` to both sides. Now plot these `x`-intercept points on the graph.
Rotate `180°` and sketch the original graph around the point `(0,0)` and going through `x`-intercept `x=1`.
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Question 3 of 5
3. Question
Given `y=x^3+2x^2-3x`.
Sketch `y=-f(-x)`
Correct
Great Work!
Incorrect
A rotation of `180` degrees (about the origin) is found when `y=f(x)` is transformed to `y=-f(-x)`.To be able to sketch the new function after the rotation of `180` degrees, find `y=-f(-x)`.Transform `y=f(x)` into `y=-f(-x)`. Simplify inside the square brackets first.`f(x)=` `x^3+2x^2-3x` `-f(-x)=` `color(red)(-)[(color(blue)(-x))^3+2(color(blue)(-x))^2-3(color(blue)(-x))]` `=` `color(red)(-)[-x^3+2x^2+3x]` `=` `x^3-2x^2-3x` Now factor `-f(-x)=x^3-2x^2-3x`.`-f(-x)=` `x^3-2x^2-3x` Remove an `x` from each term. `=` `x(x^2-2x-3)` Factor inside the brackets `=` `x(x-3)(x+1)` Set `x`, `x-3`, and `x+1` equal to zero and solve for `x`. This will give you the rotated `x`-intercepts.`x=` `0` This is the first point. `x-3=` `0` This is the second point. Simplify. `x=` `3` `x+1=` `0` This is the third point. Simplify. `x=` `-1` Now plot these `x`-intercept points on the graph.
Rotate `180°` and sketch the original graph around the point `(0,0)` and going through `x`-intercept `x=0`, `x`-intercept `x=3` and `x`-intercept `x=-1`.
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Question 4 of 5
4. Question
Rotate `y=sqrt(x)` by `180` degrees about the origin.
Correct
Great Work!
Incorrect
A rotation of `180` degrees (about the origin) is found when `y=f(x)` is transformed to `y=-f(-x)`.To be able to sketch the new function after the rotation of `180` degrees, find `y=-f(-x)`.`f(x)=` `sqrt(x)` `-f(-x)=` `color(red)(-) [sqrt(color(blue)(-x)) ]` Transform `y=f(x)` into `y=-f(-x)`. Simplify inside the square brackets first. `=` `color(red)(-)[sqrt(-x)]` `=` `-sqrt(-x)` Set `-sqrt(-x)` equal to zero and solve for `x`. This will give you the rotated `x`-intercepts.`-sqrt(-x)=` `0` Take the square of both sides `-x=` `0` Simplify. `x=` `0` Now plot this ‘ x’-intercept point on the graph.
Rotate `180°` and sketch the original graph around the point `(0,0)` and going through `x`-intercept `x=0`.
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Question 5 of 5
5. Question
Given `y=(x-1)^3`.
Sketch `y=-f(-x)`
Correct
Great Work!
Incorrect
A rotation of `180` degrees (about the origin) is found when `y=f(x)` is transformed to `y=-f(-x)`.To be able to sketch the new function after the rotation of `180` degrees, find `y=-f(-x)`.`f(x)=` `(x-1)^3` `-f(-x)=` `color(red)(-)(color(blue)(-x)-1)^3` Transform `y=f(x)` into `y=-f(-x)`. `=` `-(-x-1)^3` Set `-(-x-1)^3` equal to zero and solve for `x`. This will give you the rotated `x`-intercepts.`-(-x-1)=` `0` Taking the cube root of both sides. `x+1=` `0` Distributing the negative sign `x=` `-1` Now plot these `x`-intercept points on the graph.
Rotate `180°` and sketch the original graph around the point `(0,0)` and going through `x`-intercept `x=-1`.
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