Given the point (2,3), perform or determine the following:

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

Determine the abcissa for (2,3)

The abcissa is the absolute value of the x-coordinate, or perpendicular distance to the y-axis

Abcissa = |2| =

Determine the ordinate for (2,3)

The ordinate is the absolute value of the y-coordinate, or perpendicular distance to the x-axis

Ordinate = |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)

Since 2>0 and 3>0, (2,3) is in Quadrant I

Convert the point (2,3°) from polar coordinates to Cartesian (rectangular) coordinates

Polar Coordinates are denoted as (r,θ)

Cartesian Coordinates are denoted as (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(0.71),2(0.196))

(rcosθ,rsinθ) =

Therefore, (2,3°) in polar coordinates equals

Since 1.9973>0 and 0.1047>0, (1.9973,0.1047) is in Quadrant I

Convert the Cartesian point (2,3) to polar coordinates

Cartesian Coordinates are denoted as (x,y)

Polar Coordinates are denoted as (r,θ)

(x,y) = (2,3)

r = ±√x

r = ±√2

r = ±√4 + 9

r = ±√13

r =

θ = tan

θ = tan

θ = tan

θ

Angle in Degrees = | Angle in Radians * 180 |

π |

θ_{degrees} = | 0.33 * 180 |

π |

θ_{degrees} = | 452 |

π |

θ

Therefore, (2,3) in Cartesian coordinates equals

Since 2>0 and 3>0, (2,3) is in Quadrant I

Show various equivalent coordinates for the polar coordinate point (2,3°)

(2,3° + 360°)

(2,363°)

(2,3° + 360°)

(2,723°)

(2,3° + 360°)

(2,1083°)

(-1 * 2,3° + 180°)

(-2,183°)

(-1 * 2,3° - 180°)

(-2,-177°)

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

Therefore, our symmetric point with respect to the origin = (-2, -3)

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

Therefore, our symmetric point with respect to the x-axis = (2, -3)

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

Therefore, our symmetric point with respect to the y-axis = (-2, 3)

Take (2, 3) and rotate 90 degrees denoted as R

The formula for rotating a point 90° is R

R

R

Take (2, 3) and rotate 180 degrees denoted as R

The formula for rotating a point 180° is R

R

R

Take (2, 3) and rotate 270 degrees denoted as R

The formula for rotating a point 270° is R

R

R

Take (2, 3) and reflect over the origin axis denoted as r

The formula for reflecting a point over the origin is r

r

r

Take (2, 3) and reflect over the y-axis axis denoted as r

The formula for reflecting a point over the y-axis is r

r

r

Take (2, 3) and reflect over the x-axis axis denoted as r

The formula for reflecting a point over the x-axis is r

r

r