Ordered Pair

Enter Ordered Pair ex. (1,2)

Given the point (1,4), 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 (1,4)

Abcissa = absolute value of x-value
Perpendicular distance to the y-axis

Abcissa = |1| = 1

Determine the ordinate for (1,4)

Ordinate = absolute value of y-value
Perpendicular distance to the x-axis

Ordinate = |4| = 4

Evaluate the ordered pair (1,4)

We start at the coordinates (0,0)

Since our x coordinate of 1 is positive
We move up on the graph 1 space(s)

Since our y coordinate of 4 is positive
We move right on the graph 4 space(s)

Determine the quadrant for (1,4)

Since 1>0 and 4>0
(1,4) is in Quadrant I

Convert (1,4) 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) = (1,4°)

(rcosθ,rsinθ) = (1cos(4),1sin(4))

(rcosθ,rsinθ) = (1(0.99756405026539),1(0.069756473664546))

(rcosθ,rsinθ) = (0.9976,0.0698)

(1,4°) = (0.9976,0.0698)

Determine the quadrant for (0.9976,0.0698)

Since 0.9976>0 and 0.0698>0
(0.9976,0.0698) is in Quadrant I

Convert (1,4) to polar coordinates:

Cartesian Coordinates are (x,y)

Polar Coordinates are (r,θ)

(x,y) = (1,4)

Transform r:

r = ±√x² + y²

r = ±√1² + 4²

r = ±√1 + 16

r = ±√17

r = ±4.1231056256177

Transform θ

θ = tan^-1(y/x)

θ = tan^-1(4/1)

θ = tan^-1(4)

θ_radians = 1.325817663668

Convert our angle to degrees

Angle in Degrees  =	Angle in Radians * 180
	π

θdegrees  =	1.325817663668 * 180
	π

θdegrees  =	238.64717946025
	π

θ_degrees = 75.96°

Therefore, (1,4) = (4.1231056256177,75.96°)

Determine the quadrant for (1,4)

Since 1>0 and 4>0
(1,4) is in Quadrant I

Show equivalent coordinates

We add 360°

(1,4° + 360°)

(1,364°)

(1,4° + 360°)

(1,724°)

(1,4° + 360°)

(1,1084°)

Method 2: -(r) + 180°

(-1 * 1,4° + 180°)

(-1,184°)

Method 3: -(r) - 180°

(-1 * 1,4° - 180°)

(-1,-176°)

Determine symmetric point

If (x,y) is symmetric to the origin:
then the point (-x,-y) is also on the graph

(-1, -4)

Determine symmetric point

If (x,y) is symmetric to the x-axis:
then the point (x, -y) is also on the graph

(1, -4)

Determine symmetric point

If (x,y) is symmetric to the y-axis:
then the point (-x, y) is also on the graph

(-1, 4)

Rotate 90 Degrees

Take (1, 4) and rotate 90 degrees
We call this R_90°

The formula for rotating a point 90° is:
R_90°(x, y) = (-y, x)

R_90°(1, 4) = (-(4), 1)

R_90°(1, 4) = (-4, 1)

Rotate 180 Degrees

Take (1, 4) and rotate 180 degrees
We call this R_180°

The formula for rotating a point 180° is:
R_180°(x, y) = (-x, -y)

R_180°(1, 4) = (-(1), -(4))

R_180°(1, 4) = (-1, -4)

Rotate 270 Degrees

Take (1, 4) and rotate 270 degrees
We call this R_270°

The formula for rotating a point 270° is:
R_270°(x, y) = (y, -x)

R_270°(1, 4) = (4, -(1))

R_270°(1, 4) = (4, -1)

Reflect over the origin

Take (1, 4) and reflect over the origin
We call this r_origin

Formula for reflecting over the origin is:
r_origin(x, y) = (-x, -y)

r_origin(1, 4) = (-(1), -(4))

r_origin(1, 4) = (-1, -4)

Reflect over the y-axis

Take (1, 4) and reflect over the y-axis
We call this r_y-axis

Formula for reflecting over the y-axis is:
r_y-axis(x, y) = (-x, y)

r_y-axis(1, 4) = (-(1), 4)

r_y-axis(1, 4) = (-1, 4)

Reflect over the x-axis

Take (1, 4) and reflect over the x-axis
We call this r_x-axis

Formula for reflecting over the x-axis is:
r_x-axis(x, y) = (x, -y)

r_x-axis(1, 4) = (1, -(4))

r_x-axis(1, 4) = (1, -4)

Final Answer