Transcript Chapter 0 – Section 07 - Dr. Abdullah Almutairi

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Precalculus Review
Copyright © Cengage Learning. All rights reserved.
0.7
The Coordinate Plane
Copyright © Cengage Learning. All rights reserved.
The Coordinate Plane
The xy-plane shown in Figure 2, is nothing more than a
very large—in fact, infinitely large—flat surface.
Figure 2
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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.
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Example 1 – Coordinates of Points
a. Find the coordinates of the indicated points. (See Figure 3.
The grid lines are placed at intervals of one unit.)
Figure 3
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)
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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)
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Example 1(b) – Solution
cont’d
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.
Figure 4
To locate C, we move 3 units right and 2.5 down.
We locate the remaining points in a similar way.
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The Graph of an Equation
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The Graph of an Equation
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, 2x2 – y = 0, y = 3x2 + 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.
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Example 2 – Graph of an Equation
Obtain the graph of the equation y – x2 = 0.
Solution:
We can solve the equation for y to obtain y = x2.
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:
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Example 2 – Solution
cont’d
Plotting these points (x, y) gives the following picture (left
side of Figure 5), suggesting the graph on the right in
Figure 5.
Figure 5
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Distance
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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(x1, y1) and Q(x2, y2) is
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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 = (x2 – x1)2 + (y2 – y1)2.
Taking square roots (d is a distance, so we take the
positive square root), we get the distance formula.
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Distance
Notice that if we switch x1 with x2 or y1 with y2, 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.
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Distance
We get the following equation for the circle centered at the
origin with radius r :
Distance from the origin = r.
Squaring both sides gives the following equation:
Equation of the Circle of Radius r Centered at the Origin
x2 + y2 = r2
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Distance
Quick Example
The circle of radius 1 centered at the origin has equation
x2 + y2 = 1.
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