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Math 150 Lecture Notes
The Cartesian Coordinate System
The Coordinate Plane
The Cartesian Coordinate System is formed by two perpendicular number lines that intersect at 0 on each line. The horizontal axis is the x-axis; the vertical axis is the y-axis. The point of intersection is the origin. The two axes divide the plan into four quadrants. Any point P in the coordinate plane can be described by a unique ordered pair of numbers (a, b), for which a is the x-coordinate or abscissa and b is the y-coordinate or ordinate.
Distance and Midpoint
The distance between two points, A(x 1 , y 1 ) and B(x 2 , y 2 ) is d(A, B) = ( x 2 − x 1 )^2 +(y 2 −y 1 )^2.
The midpoint of the line segment from A(x 1 , y 1 ) to B(x 2 , y 2 ) is (^)
x 1 x 2 y 1 y (^2).
Graphs of Equations in Two Variables
An equation in two variables expresses a relationship between two quantities. A point (x, y) satisfies the equation if it makes the equation true when the values for x and y are substituted into the equation.
The graph of an equation in x and y is the set of all points (x, y) in the coordinate plane that satisfy the equation.
Intercepts
The x-intercept is the point where the graph of an equation intersects the x-axis.
The y-intercept is the point where the graph of an equation intersects the y-axis.
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Circles
A circle is the set of all points P(x, y) in a plane that are the same distance r from a given point C(h, k), which is the center of the circle. The distance r is the radius of the circle.
The standard form for the equation of the circle with center (h, k) and radius r is (x – h)^2 + (y – k)^2 = r^2
If the center of the circle is the origin (0, 0), then the equation is x^2 + y^2 = r^2.
Symmetry
When a graph is symmetric with respect to the x-axis, it is a mirror image of itself across the x- axis; when y is replaced with –y in the equation, the result (after simplification) is the same equation. For every point (x, y) on the graph, the point (x, -y) is on the graph.
When a graph is symmetric with respect to the y-axis, it is a mirror image of itself across the y- axis; when x is replaced by –x in the equation, the result (after simplification) is the same equation. For every point (x, y) on the graph, the point (-x, y) is on the graph.
When a graph is symmetric with respect to the origin, it gives the same graph when it is rotated 180° about the origin or when it is reflected across the x-axis, then the y-axis (or the y- axis, then the x-axis). When x is replaced by –x and y is replaced by –y, the resulting equation is the same (after simplification). For every point (x, y) on the graph, the point (-x, -y) is on the graph.
