Lecture on Section 1.9: Coordinate Plane, Graphs of Equations, and Circles

1. Coordinate Plane

Definition: A graphing plane for lines, points, functions, and equations.
Axes:
- X-axis: Horizontal line where x-values are plotted.
- Y-axis: Vertical line where y-values are plotted.
Quadrants: The plane is divided into four quadrants.
Points:
- Identified with ordered pairs (x, y).
- Example: Point P (A, B) is where A is the x-coordinate and B is the y-coordinate.

Distance Formula:
- Used to calculate the distance between two points.
- Formula: ( D(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
Midpoint Formula:
- Used to find the middle point between two points.
- Formula: ( M(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) )

Definition: Expresses a relationship between x and y.
Graph: Picture of all possible solutions.
Intercepts:
- X-intercepts: Where the graph crosses the x-axis.
  - Found by setting y = 0.
- Y-intercepts: Where the graph crosses the y-axis.
  - Found by setting x = 0.

Types of Symmetry:
- X-axis symmetry: Replace y with -y.
- Y-axis symmetry: Replace x with -x.
- Origin symmetry: Replace both x with -x and y with -y.

Equation of a Circle:
- Standard form: ( (x - h)^2 + (y - k)^2 = r^2 )
- Center: ( (h, k) )
- Radius: ( r )
Completing the Square:
- Used to rewrite equations into the perfect square form.
- For expression ( x^2 + bx ), add ( \left(\frac{b}{2}\right)^2 ).

Example 1:
- Calculating the distance and midpoint between points P(1, -2) and A(3, 2).
- Distance = ( \sqrt{20} ) or approximately 4.47.
- Midpoint = (2, 0).
Example 2:
- Graph the equation ( y = x^2 - 4 ).
- Find x and y-intercepts: ((-2,0)), ((2,0)) for x-intercepts; ((0,-4)) for y-intercept.
Example 3:
- Test ( y = x^2 - 4 ) for symmetry.
- Result: Symmetric to the y-axis.
Example 4:
- Rewrite ( x^2 + y^2 + 10x - 6y + 33 = 0 ) in standard form and find center & radius.
- Result: Center is (-5, 3), Radius is 1.