1. Copyright © Cengage Learning. All rights reserved. 0 Precalculus Review

2. Copyright © Cengage Learning. All rights reserved. 0.7 The Coordinate Plane

3. 3 The xy-plane shown in Figure 2, is nothing more than a very large—in fact, infinitely large—flat surface. Figure 2

4. 4 The Coordinate Plane The purpose of the axes is to allow us to locate specific positions, or points, on the plane, with the use of coordinates. The way of assigning coordinates to points in the plane is often called the system of Cartesian coordinates.

5. 5 a. Find the coordinates of the indicated points. (See Figure 3. The grid lines are placed at intervals of one unit.) b. Locate the following points in the xy-plane. A(2, 3), B(–4, 2), C(3, –2.5), D(0, –3), E(3.5, 0), F(–2.5, –1.5) Example 1 – Coordinates of Points Figure 3

6. 6 Example 1(a) – Solution Taking them in alphabetical order, we start with the origin O. This point has height zero and is also zero units to the right of the y-axis, so its coordinates are (0, 0). Turning to P, dropping a vertical line gives x = 2 and extending a horizontal line gives y = 4. Thus, P has coordinates (2, 4). For practice, determine the coordinates of the remaining points, and check your work against the list that follows: Q(–1, 3), R(–4, –3), S(–3, 3), T(1, 0), U(2.5, –1.5)

7. 7 Example 1(b) – Solution In order to locate the given points, we start at the origin (0, 0), and proceed as follows. (See Figure 4.) To locate A, we move 2 units to the right and 3 up, as shown. To locate B, we move –4 units to the right (that is, 4 to the left) and 2 up, as shown. To locate C, we move 3 units right and 2.5 down. We locate the remaining points in a similar way. cont’d Figure 4

8. 8 The Graph of an Equation

9. 9 One of the more surprising developments of mathematics was the realization that equations, which are algebraic objects, can be represented by graphs, which are geometric objects. The kinds of equations that we have in mind are equations in x and y, such as y = 4x – 1, 2x 2 – y = 0, y = 3x 2 + 1, y =. The graph of an equation in the two variables x and y consists of all points (x, y) in the plane whose coordinates are solutions of the equation.

10. 10 Obtain the graph of the equation y – x 2 = 0. Solution: We can solve the equation for y to obtain y = x 2. Solutions can then be obtained by choosing values for x and then computing y by squaring the value of x, as shown in the following table: Example 2 – Graph of an Equation

11. 11 Example 2 – Solution Plotting these points (x, y) gives the following picture (left side of Figure 5), suggesting the graph on the right in Figure 5. cont’d Figure 5

12. 12 Distance

13. 13 Distance The distance between two points in the xy-plane can be expressed as a function of their coordinates, as follows: Distance Formula The distance between the points P(x 1, y 1 ) and Q(x 2, y 2 ) is

14. 14 Distance Derivation The distance d is shown in the figure below. By the Pythagorean theorem applied to the right triangle shown, we get d 2 = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2. Taking square roots (d is a distance, so we take the positive square root), we get the distance formula.

15. 15 Distance Notice that if we switch x 1 with x 2 or y 1 with y 2, we get the same result. Quick Example The distance between the points (3, –2) and (–1, 1) is The set of all points (x, y) whose distance from the origin (0, 0) is a fixed quantity r is a circle centered at the origin with radius r.

16. 16 Distance We get the following equation for the circle centered at the origin with radius r : Squaring both sides gives the following equation: Equation of the Circle of Radius r Centered at the Origin x 2 + y 2 = r 2 Distance from the origin = r.

17. 17 Distance Quick Example The circle of radius 1 centered at the origin has equation x 2 + y 2 = 1.