**Section Objectives**

- Graph two-variable equations in the rectangular coordinate system.
- Find the distance between two points in the rectangular coordinate system.
- Determine the midpoint of two points.
- Use the standard form equation of a circle.
- Graph circles.
- Gain familiarity with some basic graphs.

The ** rectangular (or Cartesian) coordinate system** consists of two perpendicular number lines intersecting at their zero marks. The number lines are called axes---the horizontal axis is typically called the -axis, while the vertical axis is the -axis. The horizontal axis increases to the right; the vertical axis increases upward.

A

A point is a ** solution** of a two-variable equation if substitution of the coordinates for the corresponding variables yields a true statement.

The ** graph** of a two-variable equation is the collection of all points that satisfy the equation.

For now, we will determine the graph of a two-variable equation by finding several solutions, plotting them, hoping they provide a representative sample, and generalizing to the overall graph from our sample.

- Find five (5) points that satisfy the equation . Plot them and sketch the graph.

- Notice that the equation above is a linear equation (in two variables). Here is another: . Find some points that are solutions, and sketch the graph.

- Find five (5) points that satisfy the equation . Plot them and sketch the graph.

- Based on the previous example, can you guess the graph of .

- Find five (5) points that satisfy the equation . Plot them and sketch the graph.

**Important point: The graphs of linear equations, , and are considered basic graphs. These are graphs you should quickly become familiar with.**

Think about two distinct points in the rectangular coordinate system. Imagine connecting them with a line segment. The ** midpoint** of the segment is determined by finding the average of the coordinates:

The midpoint of the segment joining and is given by .

The length of the line segment or ** distance between the points** can be obtained from the Pythagorean theorem:

Distance from to .

- Find the distance from to . Write your answer in exact form.

- Determine the distance from to . Round to the nearest hundredth.

- In each example above, determine the midpoint of the given points.

A ** circle** is the set of all points in the plane (rectangular coordinate system) whose distance from a fixed point, called the

It follows from the distance formula that the circle of radius centered at can be described by the ** standard form equation**:

.

- Find the standard form equation of the circle with center and radius .

- Graph the circle described by .

- The equation describes a circle. Write the equation in standard form and determine the center and radius of the circle.