Download P.3 The Cartesian Plane and Graphs of Equations

Section P.3
9
The Cartesian Plane and Graphs of Equations
Course Number
Section P.3 The Cartesian Plane and Graphs of Equations
Instructor
Objective: In this lesson you learned how to plot points in the
coordinate plane, use the Distance and Midpoint Formulas,
and sketch graphs of equations.
Important Vocabulary
Date
Define each term or concept.
Rectangular coordinate system A plane (Cartesian plane) used to graphically
represent ordered pairs of real numbers.
Ordered pair Two real numbers x and y, written (x, y), which represent a point in the
Cartesian plane.
Graph of an equation The set of all points that are solutions of the equation.
Intercepts The points at which a graph intersects the x- or y-axis.
Symmetry If a graph is folded along a dividing line and the portion of the graph on
one side of the dividing line coincides with the portion of the graph on the other side
of the dividing line, then the graph is said to have symmetry.
I. The Cartesian Plane (Pages 25−26)
On the Cartesian plane, the horizontal real number line is usually
called the
, and the vertical real number line is
x-axis
usually called the
y-axis
of intersection
. The origin is the point
of these two axes, and the two axes divide
the plane into four parts called
quadrants
.
On the Cartesian plane shown below, label the x-axis, the y-axis,
the origin, Quadrant I, Quadrant II, Quadrant III, and Quadrant
IV.
y-axis
5
Quadrant II
1
-5
-3
Quadrant III
Quadrant I
3
-1
-1
-3
origin
1
3
5
x-axis
Quadrant IV
-5
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.
What you should learn
How to plot points in the
Cartesian plane
10
Chapter P
To sketch a scatter plot of paired data given in a table, . . .
5
Prerequisites
y
represent each pair of values by an ordered pair and plot the
3
resulting points.
1
Example 1: Explain how to plot the ordered pair (3, − 2), and
then plot it on the Cartesian plane provided.
Imagine a vertical line through 3 on the x-axis and
a horizontal line through − 2 on the y-axis. The
intersection of these two lines is the point (3, − 2).
II. The Distance Formula (Page 27)
The Distance Formula states that . . .
the distance d between
the points (x1, y1) and (x2, y2) in the plane is
d = √ (x2 − x1)2 + (y2 − y1)2
.
-5
-3
-1
-1
1
3
5
x
(3, -2)
-3
-5
What you should learn
How to use the Distance
Formula to find the
distance between two
points
Example 2: Explain how to use the Distance Formula to find
the distance between the points (4, 2) and (5, − 1).
Then find the distance and round to the nearest
hundredth.
Explanations will vary. Let x1 = 4 and y1 = 2, and
let x2 = 5 and y2 = − 1. Then substitute these values
into the Distance Formula and simplify; 3.16
III. The Midpoint Formula (Page 28)
To find the midpoint of a line segment, find the
values
average
of the respective coordinates of the two
endpoints of the line segment using the Midpoint Formula.
What you should learn
How to use the Midpoint
Formula to find the
midpoint of a line
segment
The Midpoint Formula gives the midpoint of the segment
joining the points (x1, y1) and (x2, y2) as . . .
Midpoint =
 x1 + x2
y1 + y2 
  ,  

2
2

Example 3: Explain how to find the midpoint of the line segment with
endpoints at (− 8, 2) and (6, − 10). Then find the
coordinates of the midpoint.
Explanations will vary. Find the average of the
two x-coordinates and find the average of the two
y-coordinates. These averages form the
coordinates of the midpoint, (− 1, − 4).
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.
Section P.3
11
The Cartesian Plane and Graphs of Equations
IV. The Graph of an Equation (Page 29)
What you should learn
How to sketch graphs of
equations
To sketch the graph of an equation in two variables, . . .
construct a table of values that consists of several solution points
of the equation. Plot the solution points on a rectangular
5
coordinate system and connect the points with a smooth curve.
y
3
Example 4: Complete the table. Then use the resulting solution
points to sketch the graph of the equation
y = 3 − 0.5x.
x
y
−4
5
−2
4
0
3
2
2
4
1
1
-5
-3
-1
-1
1
3
-3
-5
V. Intercepts of a Graph (Page 30)
An x-intercept is written as the ordered pair
a y-intercept is written as the ordered pair
To find x-intercepts, . . .
, and
(x, 0)
(0, y)
.
What you should learn
How to find x- and
y-intercepts of graphs of
equations
let y be zero and solve the equation
for x.
To find y-intercepts, . . .
let x be zero and solve the equation
for y.
Example 5: For the equation 3 x − 4 y = 12 , find:
(a) the x-intercept(s), and (b) the y-intercept(s).
(a) (4, 0)
(b) (0, − 3)
VI. Symmetry (Pages 31−33)
The three types of symmetry that a graph can exhibit are . . .
y-axis symmetry, origin symmetry, or x-axis symmetry.
Knowing the symmetry of a graph before attempting to sketch it
is helpful because . . .
then you need only half as many
solution points to sketch the graph.
A graph is symmetric with respect to the y-axis if, whenever
(x, y) is on the graph,
(− x, y)
is also on the graph. A graph
is symmetric with respect to the x-axis if, whenever (x, y) is on
the graph,
(x, − y)
is also on the graph. A graph is
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.
What you should learn
How to use symmetry to
sketch graphs of
equations
5
x
12
Chapter P
Prerequisites
symmetric with respect to the origin if, whenever (x, y) is on the
(− x, − y)
graph,
is also on the graph.
The graph of an equation is symmetric with respect to the y-axis
if . . .
replacing x with − x yields an equivalent equation.
The graph of an equation is symmetric with respect to the x-axis
if . . .
replacing y with − y yields an equivalent equation.
35
30
The graph of an equation is symmetric with respect to the origin
if . . .
y
25
replacing x with − x and y with − y yields an equivalent
20
equation.
15
10
5
Example 6: Use intercepts and symmetry to sketch the graph
of the equation y = 2 x 2 + 2 .
0
-5
VII. Circles (Page 33)
(x − h)2 + (y − k)2 = r2
-1
-5
1
3
5
x
What you should learn
How to find equations
and sketch graphs of
circles
The standard form of the equation of a circle with center
(h, k) and radius r is
-3
.
The standard form of the equation of a circle with radius r and its
center at the origin is
x2 + y2 = r 2
.
5
Example 7: For the equation ( x + 2) + ( y − 1) = 4 , find the
center and radius of the circle and then sketch the
graph of the equation.
2
2
y
3
1
Center: (− 2, 1)
Radius: 2
-5
-3
-1
-1
1
3
5
x
-3
-5
Homework Assignment
Page(s)
Exercises
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.