Graphs
Chapter 1
Rectangular
Coordinates;
Graphing Utilities
Section 1.1
Rectangular Coordinate System
Rectangular Coordinate System

Example.
Problem: Plot the points (0,7), ({6,0),
(6,4) and ({3,{5)
Answer:
10
7.5
5
2.5
-10
-5
5
-2.5
-5
-7.5
-10
10
Rectangular Coordinate System

The points on the axes are not
considered to be in any quadrant
Quadrant II
x < 0, y > 0
Quadrant I
x > 0, y > 0
Quadrant III
x < 0, y < 0
Quadrant IV
x > 0, y < 0
Distance Formula

Theorem [Distance Formula] The
distance between two points
P1 = (x1, y1) and P2 = (x2, y2),
denoted by d(P1, P2), is
Distance Formula

Example.
Problem: Find the distance between the
points (6,4) and ({3,{5).
Answer:
Midpoint Formula

Theorem [Midpoint Formula] The
midpoint M = (x,y) of the line
segment from P1 = (x1, y1) to P2 =
(x2, y2) is
Midpoint Formula

Example.
Problem: Find the midpoint of the line
segment between the points (6,4) and
({3,{5)
Answer:
Key Points



Rectangular Coordinate System
Distance Formula
Midpoint Formula
Graphs of
Equations in Two
Variables
Section 1.2
Solutions of Equations


Solutions of an equation: Points that
make the equation true when we
substitute the appropriate numbers
for x and y
Example.
Problem: Do either of the points ({3,{10)
or (2,4) satisfy the equation y = 3x { 1?
Answer:
Graphs of Equations


Graph of an equation: Set of points in
plane whose coordinates (x, y) satisfy
the equation
To plot a graph:
List some solutions
 Connect the points
 More sophisticated methods seen later

Graphs of Equations

Example.
Problem: Graph the equation y = 3x{1
Answer:
10
5
-10
-5
5
-5
-10
10
Graphs of Equations

Example.
Problem: Graph the equation y2 = x
Answer:
4
2
2
-2
-4
4
6
8
10
Intercepts




Intercepts: Points where a graph
crosses or touches the axes, if any
x-intercepts: x-coordinates of
intercepts
y-intercepts: y-coordinates of
intercepts
May be any number of x- or yintercepts
Intercepts

Example.
Problem: Find all intercepts of the graph
Answer:
2
1
-1
1
-1
-2
2
3
4
Intercepts

Finding intercepts from an equation
To
set
 To
set

find the x-intercepts of an equation,
y=0 and solve for x
find the y-intercepts of an equation,
x=0 and solve for y
Intercepts

Example.
Problem: Find the intercepts of the
equation 4x2 + 25y2 = 100
Answer:
6
4
2
-6
-4
-2
2
-2
4
6
Symmetry



Symmetry with respect to the xaxis: If (x,y) is on the graph, then so
is (x, {y)
Symmetry with respect to the yaxis: If (x,y) is on the graph, then so
is ({x, y)
Symmetry with respect to the
origin: If (x,y) is on the graph, then
so is ({x, {y)
Symmetry and Graphs

x-axis symmetry means that the
portion of the graph below the x-axis
is a reflection of the portion above it
2
1
-1
1
-1
-2
2
3
4
Symmetry and Graphs

y-axis symmetry means that the
portion of the graph to the left of the
y-axis is a reflection of the portion to
the right of it
4
3
2
1
-2
-1
1
2
Symmetry and Graphs

Origin symmetry
Reflection across one axis, then the other
 Projection along a line through origin so
that distances from the origin are equal
 Rotation of 180± about the origin

10
7.5
5
2.5
-10
-5
5
-2.5
-5
10
Symmetry and Equations

To test an equation for
x-axis symmetry: Replace y by {y
 y-axis symmetry: Replace x by {x
 origin symmetry: Replace x by {x and y
by {y
 In each case, if an equivalent equation
results, the graph has the appropriate
symmetry

Symmetry and Equations

Example.
Problem: Test the equation
x2 {4x + y2 { 5 = 0 for symmetry
Answer:
Important Equations

y = x2
x-intercept: x = 0
 y-intercept: y = 0
 Symmetry: y-axis only

10
7.5
5
2.5
-10
-5
5
-2.5
-5
10
Important Equations

x = y2
x-intercept: x = 0
 y-intercept: y = 0
 Symmetry: x-axis only

10
7.5
5
2.5
-10
-5
5
-2.5
-5
-7.5
-10
10
Important Equations

x-intercept: x = 0
 y-intercept: y = 0
 Symmetry: None

10
7.5
5
2.5
-10
-5
5
-2.5
-5
-7.5
-10
10
Important Equations

y=x3
10
x-intercept: x = 0
 y-intercept: y = 0
 Symmetry: Origin only

7.5
5
2.5
-10
-5
5
-2.5
-5
Important Equations

y=
10
7.5
x-intercept: None
 y-intercept: None
 Symmetry: Origin only

5
2.5
-10
-5
5
-2.5
-5
Key Points







Solutions of Equations
Graphs of Equations
Intercepts
Symmetry
Symmetry and Graphs
Symmetry and Equations
Important Equations
Solving Equations in
One Variable Using
a Graphing Utility
Section 1.3
Using Zero or Root to
Approximate Solutions

Example.
Problem: Find the solutions to the
equation x3 { 6x + 3 = 0. Approximate
to two decimal places.
Answer:
Use Intersect to Solve
Equations

Example.
Problem: Find the solutions to the
equation {x4 + 3x3 + 2x2 = {2x + 1.
Approximate to two decimal places.
Answer:
Key Points


Using Zero or Root to Approximate
Solutions
Use Intersect to Solve Equations
Lines
Section 1.4
Slope of a Line

P = (x1, y1) and Q = (x2,y2) two
distinct points

P and Q define a unique line L


If x1  x2, L is nonvertical. Its slope is
defined as
x1 = x2, L is vertical. Slope is
undefined.
Slope of a Line
Slope of a Line

Interpretation of the slope of a nonvertical
line

Average rate of change of y with respect to
x, as x changes from x1 to x2
Slope of a Line

Any two distinct points serve to
compute the slope

The slope from P to Q is the same as
the slope from Q to P
Slope of a Line

Example.
Problem: Compute the slope of the line
containing the points (7,3) and ({2,{2)
Answer:
Slope of a Line

Move from left to right
Line slants upward if the slope is
positive
 Line slants downward if slope is negative
 Line is horizontal if the slope is 0


Larger magnitudes correspond to
steeper slopes
Slope of a Line
10
m 3
7.5
m 1
5
1
m
3
2.5
m 0
-10
-5
5
10
-2.5
1
m
3
-5
m
-7.5
m
-10
3
1
Slope of a Line

Example.
Problem: Draw the graph of the line
containing the point (1,5) with a slope of
10
7.5
5
Solution:
2.5
-10
-5
5
-2.5
-5
-7.5
-10
10
Equations of Lines

Theorem [Equation of a Vertical Line]
A vertical line is given by an equation
of the form
x=a
where a is the x-intercept
Equations of Lines

Example.
Problem: Find an equation of the vertical
line passing through the point ({1, 2)
Answer:
Equations of Lines

Theorem. [Equation of a Horizontal
Line]
A horizontal line is given by an
equation of the form
y=b
where b is the y-intercept
Equations of Lines

Example.
Problem: Find an equation of the
horizontal line passing through the point
({1, 2)
Answer:
Point-Slope Form of a Line

Theorem. [Point-Slope Form of an
Equation of a Line]
An equation of a nonvertical line of
slope m that contains the point
(x1, y1) is
y { y1= m(x { x1)
Point-Slope Form of a Line

Example.
Problem: Find an equation of the line with
slope
point ({1, 2)
Answer:
passing through the
Point-Slope Form of a Line

Example.
Problem: Find an equation of the line
containing the points ({1, 2) and (5,3).
Answer:
Slope-Intercept Form of a Line

Theorem. [Slope-Intercept Form of an
Equation of a Line]
An equation of a nonvertical line L
with of slope m and y-intercept b
y = mx + b
Slope-Intercept Form of a Line

Example.
Problem: Find the slope-intercept form of
the line in the graph
10
7.5
Answer:
5
2.5
-10
-5
5
-2.5
-5
10
General Form of a Line




General form of a line L:
Ax + By = C
A, B and C are real numbers, A and B not
both 0.
Any line, vertical or nonvertical, may be
expressed in general form
The general form is not unique
Any equation which is equivalent to the
general form of a line is called a linear
equation
Parallel Lines


Parallel Lines: Two lines which do
not intersect
Theorem. [Criterion for Parallel Lines]
Two nonvertical lines are parallel if
and only if their slopes are equal and
they have different y-intercepts.
Parallel Lines

Example.
Problem: Find the line passing through
the point (1, {2) which is parallel to the
line y = 3x + 2
Answer:
Perpendicular Lines

Perpendicular lines: Two lines that
intersect at a right angle
Perpendicular Lines


Theorem. [Criterion for Perpendicular
Lines] Two nonvertical lines are
perpendicular if and only if the
product of their slopes is {1.
The slopes of perpendicular lines are
negative reciprocals of each other
Perpendicular Lines

Example.
Problem: Find the line passing through
the point (1, {2) which is parallel to the
line y = 3x + 2
Answer:
Key Points







Slope of a Line
Equations of Lines
Point-Slope Form of a Line
Slope-Intercept Form of a Line
General Form of a Line
Parallel Lines
Perpendicular Lines
Circles
Section 1.5
Circles



Circle: Set of points in xy-plane that
are a fixed distance r from a fixed
point (h,k)
r is the radius
(h,k) is the center of the circle
Standard Form of a Circle


Standard form of an equation of a
circle with radius r and center (h, k)
is
(x{h)2 + (y{k)2 = r2
Standard form of an equation
centered at the origin with radius r is
x 2 + y 2 = r2
Standard Form of a Circle

Example.
Problem: Graph the equation
(x{2)2 + (y+4)2 = 9
Answer:
8
6
4
2
-7.5
-5
-2.5
2.5
-2
-4
-6
-8
5
7.5
Unit Circle


Unit Circle: Radius r = 1 centered at
the origin
Has equation x2 + y2 = 1
2
1.5
1
0.5
-2
-1
1
-0.5
-1
-1.5
2
General Form of a Circle


General form of the equation of a
circle
x2 + y2 + ax + by + c = 0
if this equation has a circle for a
graph
If given a general form, complete the
square to put it in standard form
General Form of a Circle

Example.
Problem: Find the center and radius of the
circle with equation
x2 + y2 + 6x { 2y + 6 = 0
Answer:
Key Points




Circles
Standard Form of a Circle
Unit Circle
General Form of a Circle