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Question 1 of 5
Given y=(x-2)2-1.
Sketch y=-f(-x)
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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)= |
|
-[((-x-2)2–1] |
Transform y=f(x) into y=-f(-x). Simplify inside the square brackets first. |
= |
|
-[(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= |
|
±1 |
Taking the square root of both sides. |
x= |
|
±1–2 |
Subtracting 2to 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
Given y=(x+1)2.
Sketch y=-f(-x)
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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)= |
|
-[((-x+1)2] |
Transform y=f(x) into y=-f(-x). Simplify inside the square brackets first. |
= |
|
-[(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
Given y=x3+2x2-3x.
Sketch y=-f(-x)
Incorrect
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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)= |
|
x3+2x2-3x |
-f(-x)= |
|
-[(-x)3+2(-x)2-3(-x)] |
= |
|
-[-x3+2x2+3x] |
|
= |
|
x3-2x2-3x |
|
Now factor -f(-x)=x3-2x2-3x.
-f(-x)= |
|
x3-2x2-3x |
Remove an x from each term. |
= |
|
x(x2-2x-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= |
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3 |
|
x+1= |
|
0 |
This is the third point. Simplify. |
x= |
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-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
Rotate y=√x by 180 degrees about the origin.
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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 |
-f(-x)= |
|
-[√-x] |
Transform y=f(x) into y=-f(-x). Simplify inside the square brackets first. |
= |
|
-[√-x] |
|
= |
|
-√-x |
|
Set -√-x equal to zero and solve for x. This will give you the rotated x-intercepts.
-√-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
Given y=(x-1)3.
Sketch y=-f(-x)
Incorrect
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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)= |
|
-(-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= |
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-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.