Given the point (2,3), determine:
Abcissa, Ordinate, Ordered Pair Detail, Quadrant, Polar to Cartesian, Cartesian to Polar, Equivalent Coordinates, Symmetric Points about the origin, Symmetric Points about the x-axis, Symmetric Points about the y-axis, Rotate 90 Degrees, Rotate 180 Degrees, Rotate 270 Degrees, Reflect over origin, Reflect over y-axis, Reflect over x-axis
Plot this on the Cartesian Graph:
Determine the abcissa for (2,3)
Abcissa = absolute value of x-value
Perpendicular distance to the y-axis
Abcissa = |2| = 2
Determine the ordinate for (2,3)
Ordinate = absolute value of y-value
Perpendicular distance to the x-axis
Ordinate = |3| = 3
Evaluate the ordered pair (2,3)
We start at the coordinates (0,0)
Since our x coordinate of 2 is positive
We move up on the graph 2 space(s)
Since our y coordinate of 3 is positive
We move right on the graph 3 space(s)
Determine the quadrant for (2,3)
Since 2>0 and 3>0(2,3) is in Quadrant I
Convert (2,3) to cartesian coordinates:
Polar Coordinates are (r,θ)
Cartesian Coordinates are (x,y)
Polar to Cartesian Transformation is
(r,θ) → (x,y) = (rcosθ,rsinθ)
(r,θ) = (2,3°)
(rcosθ,rsinθ) = (2cos(3),2sin(3))
(rcosθ,rsinθ) = (2(0.99862953475771),2(0.052335956183196))
(rcosθ,rsinθ) = (1.9973,0.1047)
(2,3°) = (1.9973,0.1047)
Determine the quadrant for (1.9973,0.1047)
Since 1.9973>0 and 0.1047>0(1.9973,0.1047) is in Quadrant I
Convert (2,3) to polar coordinates:
Cartesian Coordinates are (x,y)
Polar Coordinates are (r,θ)
(x,y) = (2,3)
Transform r:
r = ±√x2 + y2
r = ±√22 + 32
r = ±√4 + 9
r = ±√13
r = ±3.605551275464
Transform θ
θ = tan-1(y/x)
θ = tan-1(3/2)
θ = tan-1(1.5)
θradians = 0.98279372324733
Convert our angle to degrees
| Angle in Degrees = | Angle in Radians * 180 |
| π |
| θdegrees = | 0.98279372324733 * 180 |
| π |
| θdegrees = | 176.90287018452 |
| π |
θdegrees = 56.31°
Therefore, (2,3) = (3.605551275464,56.31°)
Determine the quadrant for (2,3)
Since 2>0 and 3>0(2,3) is in Quadrant I
Show equivalent coordinates
We add 360°
(2,3° + 360°)
(2,363°)
(2,3° + 360°)
(2,723°)
(2,3° + 360°)
(2,1083°)
Method 2: -(r) + 180°
(-1 * 2,3° + 180°)
(-2,183°)
Method 3: -(r) - 180°
(-1 * 2,3° - 180°)
(-2,-177°)
Determine symmetric point
If (x,y) is symmetric to the origin:
then the point (-x,-y) is also on the graph
(-2, -3)
Determine symmetric point
If (x,y) is symmetric to the x-axis:
then the point (x, -y) is also on the graph
(2, -3)
Determine symmetric point
If (x,y) is symmetric to the y-axis:
then the point (-x, y) is also on the graph
(-2, 3)
Rotate 90 Degrees
Take (2, 3) and rotate 90 degrees
We call this R90°
The formula for rotating a point 90° is:
R90°(x, y) = (-y, x)
R90°(2, 3) = (-(3), 2)
R90°(2, 3) = (-3, 2)
Rotate 180 Degrees
Take (2, 3) and rotate 180 degrees
We call this R180°
The formula for rotating a point 180° is:
R180°(x, y) = (-x, -y)
R180°(2, 3) = (-(2), -(3))
R180°(2, 3) = (-2, -3)
Rotate 270 Degrees
Take (2, 3) and rotate 270 degrees
We call this R270°
The formula for rotating a point 270° is:
R270°(x, y) = (y, -x)
R270°(2, 3) = (3, -(2))
R270°(2, 3) = (3, -2)
Reflect over the origin
Take (2, 3) and reflect over the origin
We call this rorigin
Formula for reflecting over the origin is:
rorigin(x, y) = (-x, -y)
rorigin(2, 3) = (-(2), -(3))
rorigin(2, 3) = (-2, -3)
Reflect over the y-axis
Take (2, 3) and reflect over the y-axis
We call this ry-axis
Formula for reflecting over the y-axis is:
ry-axis(x, y) = (-x, y)
ry-axis(2, 3) = (-(2), 3)
ry-axis(2, 3) = (-2, 3)
Reflect over the x-axis
Take (2, 3) and reflect over the x-axis
We call this rx-axis
Formula for reflecting over the x-axis is:
rx-axis(x, y) = (x, -y)
rx-axis(2, 3) = (2, -(3))
rx-axis(2, 3) = (2, -3)
Final Answer
Ordinate = |3| = 3
Quadrant = I
Quadrant = I
r = ±3.605551275464
θradians = 0.98279372324733
(2,3) = (3.605551275464,56.31°)
Quadrant = I
