Cartesian coordinates

Examples of Cartesian coordinates in the following topics:

• The Cartesian System

  • The Cartesian coordinate system is used to visualize points on a graph by showing the points' distances from two axes.
  • A Cartesian coordinate system is used to graph points.
  • The Cartesian coordinate system is broken into four quadrants by the two axes.
  • The four quadrants of theCartesian coordinate system.
  • The Cartesian coordinate system with 4 points plotted, including the origin, at $(0,0)$.

• Converting Between Polar and Cartesian Coordinates

  • Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry.
  • When given a set of polar coordinates, we may need to convert them to rectangular coordinates.
  • A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.
  • A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.
  • Derive and use the formulae for converting between Polar and Cartesian coordinates

• Other Curves in Polar Coordinates

  • Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates.
  • To graph in the rectangular coordinate system we construct a table of $x$ and $y$  values.
  • To graph in the polar coordinate system we construct a table of $r$ and $\theta$ values.

• Introduction to Circles

  • Since we know a circle is the set of points a fixed distance from a center point, let's look at how we can construct a circle in a Cartesian coordinate plane with variables $x$ and $y$.
  • The circle with center $\left(a,b\right)$ is graphed in the Cartesian plane.

• Introduction to the Polar Coordinate System

  • The polar coordinate system is an alternate coordinate system where the two variables are $r$ and $\theta$, instead of $x$ and $y$.
  • When we think about plotting points in the plane, we usually think of rectangular coordinates $(x,y)$ in the Cartesian coordinate plane.
  • The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis.
  • The radial coordinate is often denoted by $r$ or $ρ$ , and the angular coordinate by $ϕ$, $θ$, or $t$.
  • In green, the point with radial coordinate $3$ and angular coordinate $60$ degrees or $(3,60^{\circ})$.

• Standard Equations of Hyperbolas

  • If the transverse axis of any hyperbola is aligned with the x-axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as:

• Equations in Two Variables

  • Equations in two variables often express a relationship between the variables $x$ and $y$, which correspond to Cartesian coordinates.

• Functions and Their Notation

  • If both the input and output are real numbers then the ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function.

• Parts of a Parabola

  • Parabolas have several recognizable features that characterize their shape and placement on the Cartesian plane.
  • Due to the fact that parabolas are symmetric, the $x$-coordinate of the vertex is exactly in the middle of the $x$-coordinates of the two roots.

• Graphing Equations

  • We will use the Cartesian plane, in which the $x$-axis is a horizontal line and the $y$-axis is a vertical line.
  • After creating a few $x$ and $y$ ordered pairs, we will plot them on the Cartesian plane and connect the points.