consider the graph of the equation xy=9. rotate the graph 45 degrees clockwise about the center (0,0). what is the equation of the rotated graph?
the rotation matrix is
⎡cos45 -sin45⎤
⎣sin45 cos45 ⎦
⎡√2/2 - √2/2⎤
=⎣√2/2 √2/2⎦
multiply this by
⎡x⎤
⎣y⎦
gave me
x' = √2/2 x - √2/2 y
y' = √2/2 x + √2/2 y
subbing that into xy=9
(√2/2 x - √2/2 y)(√2/2 x + √2/2 y) = 9
2x^2/4 - 2y^2/4 = 9
x^2 - y^2 = 18
To rotate the graph 45 degrees clockwise about the origin (0,0), we can use the rotation matrix:
[R] = [cosθ -sinθ]
[sinθ cosθ]
where θ is the angle of rotation, in this case, 45 degrees.
Let's substitute the values into the rotation matrix:
[R] = [cos45 -sin45]
[sin45 cos45]
Simplifying the values:
[R] = [√2/2 -(√2/2)]
[√2/2 √2/2]
Now, let's apply the rotation transformation to the original equation xy = 9:
(x', y') = [R] * (x, y)
Substituting the rotation matrix:
(x', y') = [√2/2 -(√2/2)] * (x, y)
[√2/2 √2/2]
Multiplying the rotation matrix by (x, y):
x' = (√2/2)x - (√2/2)y
y' = (√2/2)x + (√2/2)y
Now, let's express y' in terms of x' using the original equation xy = 9:
xy = 9
y = 9/x
Substituting y in terms of x:
y' = (√2/2)x + (√2/2)(9/x)
Simplifying:
y' = (√2/2)x + (√2/2)(9/x)
y' = (√2/2)x + (9√2/2x)
y' = (√2/2)(x + 9)
Therefore, the equation of the rotated graph is: y' = (√2/2)(x + 9)
To rotate the graph 45 degrees clockwise about the origin (0,0), we can use a rotation matrix.
The rotation matrix for a clockwise rotation by angle θ is given by:
| cos(θ) -sin(θ) |
| sin(θ) cos(θ) |
In this case, since we want to rotate the graph 45 degrees clockwise, we substitute θ = -45 degrees or -π/4 radians into the rotation matrix.
The general equation of the graph is xy = 9. To rotate the graph, we need to substitute new variables, say X and Y, into the equation. Let's perform a change of variables:
X = x * cos(θ) - y * sin(θ)
Y = x * sin(θ) + y * cos(θ)
Substituting θ = -45 degrees or -π/4 radians:
X = x * cos(-π/4) - y * sin(-π/4)
Y = x * sin(-π/4) + y * cos(-π/4)
Simplifying the equations:
X = (x√2)/2 + (y√2)/2
Y = -(x√2)/2 + (y√2)/2
Now, substitute X and Y back into the equation xy = 9:
((x√2)/2 + (y√2)/2) * (-(x√2)/2 + (y√2)/2) = 9
Simplifying further:
(-2x^2 + 2y^2)/4 = 9
Multiply both sides by 4:
-2x^2 + 2y^2 = 36
Simplify the equation and divide by 2:
x^2 - y^2 = -18
So, the equation of the rotated graph is x^2 - y^2 = -18.