Ordered Pair

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 the point (1,4°) from
polar to Cartesian

The formula for this is below:

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

Cartesian Coordinates are denoted as (x,y)
Polar Coordinates are denoted as (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)

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)

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)

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)

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)

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)

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

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