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Rectangular Coordinates
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:
- The origin has a value of 0 on both the x-axis and the y-axis
- Points on the x-axis to the right of the origin are associated with positive real numbers and points to the left of the origin are associated with negative real numbers. Similarly, points on the y-axis above the origin are associated with positive real numbers and points below the origin are associated with negative real numbers.
Warning: (a, b) can mean a point or an interval. Note:
- The graph of y = 0 is the x-axis.
- The graph of x = 0 is the y-axis.
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
(1 โ 2)^2 + (3 โ 5)^2 =
12 + 2^2 =
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
Graphs of Equations
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.
- To find the x-intercepts, set y to 0 and solve for x.
- To find the y-intercepts, set x to 0 and solve for y.
Examples:
- Find the x and y-intercepts of y โ x^2 + 4 = 0. x-intercepts: (set y = 0): 0 โ x^2 + 4 = 0 โโ โx^2 = โ 4 โโ x^2 = 4 โโ x = โ 2 , 2. So the x-intercepts are โ 2 , 2. y-intercepts: (set x = 0): y โ 0 + 4 = 0 โโ y = โ4. So the y-intercept = โ4.
- Find the x and y-intercepts of y โ x^2 โ 4 = 0. x-intercepts: (set y = 0): 0 โ x^2 โ 4 = 0 โโ x^2 = โ4 So x-intercepts = none. y-intercepts: (set x = 0): y โ 0 โ 4 = 0 โโ y = 4. So y-intercept = 4.
