Given the point (3,4), perform or determine the following:

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

Evaluate the ordered pair (3,4)

We start at the coordinates (0,0)

Since our x coordinate of 3 is positive, we move up on the graph 3 space(s)

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

Since 3>0 and 4>0, (3,4) is in Quadrant I

Convert the point (3,4°) 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) = (3,4°)

(rcosθ,rsinθ) = (3cos(4),3sin(4))

(rcosθ,rsinθ) = (3(0.39),3(0.546))

(rcosθ,rsinθ) =

Therefore, (3,4°) in polar coordinates equals

Since 2.9927>0 and 0.2093>0, (2.9927,0.2093) is in Quadrant I

Convert the Cartesian point (3,4) to polar coordinates

Cartesian Coordinates are denoted as (x,y)

Polar Coordinates are denoted as (r,θ)

(x,y) = (3,4)

r = ±√x

r = ±√3

r = ±√9 + 16

r = ±√25

r =

θ = tan

θ = tan

θ = tan

θ

Angle in Degrees = | Angle in Radians * 180 |

π |

θ_{degrees} = | 0.61 * 180 |

π |

θ_{degrees} = | 029 |

π |

θ

Therefore, (3,4) in Cartesian coordinates equals

Since 3>0 and 4>0, (3,4) is in Quadrant I

Show various equivalent coordinates for the polar coordinate point (3,4°)

(3,4° + 360°)

(3,364°)

(3,4° + 360°)

(3,724°)

(3,4° + 360°)

(3,1084°)

(-1 * 3,4° + 180°)

(-3,184°)

(-1 * 3,4° - 180°)

(-3,-176°)

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 = (-3, -4)

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 = (3, -4)

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 = (-3, 4)

Take (3, 4) and rotate 90 degrees denoted as R

The formula for rotating a point 90° is R

R

R

Take (3, 4) and rotate 180 degrees denoted as R

The formula for rotating a point 180° is R

R

R

Take (3, 4) and rotate 270 degrees denoted as R

The formula for rotating a point 270° is R

R

R

Take (3, 4) and reflect over the origin axis denoted as r

The formula for reflecting a point over the origin is r

r

r

Take (3, 4) 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 (3, 4) and reflect over the x-axis axis denoted as r

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

r

r