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The basics of rectangular coordinates, including how to locate points on the plane using ordered pairs, calculate distance and midpoint between two points, and graph equations. Topics include distance formula, mid-point formula, graphs of equations, x-intercepts, y-intercepts, and graphing lines.
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We begin with two real number lines located in the same plane: one horizontal and one vertical. The horizontal line is called the x-axis, the vertical line called the y-axis, and the point of intersection called the origin. We assign coordinates to every point on these number lines using a convenient scale (usually the same).
Definition: Any point on the plane can be located by using an ordered pair. The notation (a, b) is the point on the plane whose x-coordinate is a and y-coordinate is b.
x
y
a
b
(a,b)
Note:
Warning: (a, b) can mean a point or an interval. Note:
Distance Formula The distance d between points (x, y) and (a, b) is d =
(x โ a)^2 + (y โ b)^2.
Example: The distance between (1, 3) and (2, 5) is
Example: The distance between points (โ 2 , 1) and (2, 3) is
โ [2 โ (โ2)]^2 + [3 โ 1]^2 =
Mid-point Formula For any two points (x 1 , y 1 ) and (x 2 , y 2 ), the mid-point M between the two points is:
M =
( (^) x 1 +x 2 2 ,^
y 1 +y 2 2
Definition: An equation in two variables, say x and y, is a statement in which two expres- sions involving x and y are equal. Any values of x and y that result in a true statement are said to satisfy the equation or are called solutions to the equation.
Definition: The graph of an equation in two variables x and y consists of the set of points in the xy-plane whose coordinates (x, y) satisfy the equation.
Example: Determine if the points A = (2, 3), and B = (2, โ2) are on the graph of the equation 2x โ y = 6.
For the point A, check to see if x = 2, y = 3 satisfies the equation:
2(2) โ 3 = 4 โ 3 = 1 6 = 6. So A is not on the graph of the equation.
For the point B, check to see if x = 2, y = โ2 satisfies the equation:
2(2) โ (โ2) = 4 + 2 = 6. So B is on the graph.
Definition: An x-intercept is the x-coordinate of a point where the graph crosses or touches the x-axis. A y-intercept is the y-coordinate of a point where the graph crosses or touches the y-axis.
The graph crosses the x-axis when the y-coordinate is 0.
Examples: