2-L

Coordinate Geometry

➜ distance between two points
➜ the midpoint of a segment
➜ slope
➜ equations of lines
➜ circles and parabolas
➜ exercises
➜ answers explained

Of the 50 questions on the Math 1 test, approximately six are on the coordinate geometry topics specifically discussed in this section. In addition, three or four more questions that the College Board would classify under other topics, such as functions and their graphs or even probability or trigonometry, could be considered as coordinate geometry questions. So, this is an important topic that you should review thoroughly.

The coordinate plane, also called the xy-plane, is formed by two perpendicular number lines called the x-axis and y-axis, which intersect at the origin. The axes divide the plane into four quadrants, labeled I, II, III, and IV as shown in the figure below. Each point in the plane is assigned two numbers, an x-coordinate and a y-coordinate, which are written as an ordered pair, (x, y). The point (2, 3) has an x-coordinate of 2 and a y-coordinate of 3 and is located at the intersection of the vertical line that crosses the x-axis at 2 and the horizontal line that crosses the y-axis at 3.

Coordinate plane showing the four quadrants, I, II, III, IV. The original is point (0, 0). Point (5, 0) is on the x-axis. Point (0, 4) is on the y-axis. Point (-4, 0) is on the x-axis. Point (0, -3) is on the y-axis. Point (2, 3) is in quadrant I. Point (-3, -5) is in quadrant III.

Key Fact L1

The y-coordinate of every point on the x-axis is 0.
The x-coordinate of every point on the y-axis is 0.
To find the x-intercepts of a graph, replace y by 0 in the equation of the graph.
To find the y-intercepts of a graph, replace x by 0 in the equation of the graph.

EXAMPLE 1: Where does the graph of y = x² – 2x – 3 cross the x-axis and where does it cross the y-axis?

By KEY FACT L1, the x-intercepts are the points whose y-coordinates are 0. So, replace y with 0:

0 = x^{2} - 2 x - 3 \Rightarrow (x - 3) (x + 1) = 0 \Rightarrow x = - 1 or x = 3

The x-intercepts of this graph are the points (–1, 0) and (3, 0).

By KEY FACT L1, the y-intercept is the point whose x-coordinate is 0. So, replace x with 0:

y = 0^{2} - 2 (0) - 3 = - 3 \Rightarrow y = - 3

The y-intercept is the point (0, –3).

The graph of y = x² – 2x – 3 is the parabola in the figure below, on which the x- and y-intercepts are indicated.