§ 1.3
Graphing Equations
The Graph of an Equation
p 29-31
The graph of an equation in two variables is the set of
points whose coordinates satisfy the equation.
An ordered pair of real numbers (x,y)
is said to satisfy the equation when substitution of the x and y coordinates
into the equation makes it a true statement.
For example, in the equation
y = 2x + 6, the ordered pair (1,8) is a solution. When we substitute this point
the sentence reads 8 = 8, which is true.
The ordered pair (2,3) is not a solution. When we substitute this point, the
sentence reads 3 = 10, which is not true.
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 1.3
Graphing an Equation
p 29-31
EXAMPLE
Graph y = 2|x| + 1.
SOLUTION
x
y = 2|x| + 1
Ordered Pair (x,y)
-2
y = 2|-2| + 1 = 2(2) + 1 = 4 + 1 = 5
(-2,5)
-1
y = 2|-1| + 1 = 2(1) + 1 = 2 + 1 = 3
(-1,3)
0
y = 2|0| + 1 = 2(0) + 1 = 0 + 1 = 1
(0,1)
1
y = 2|1| + 1 = 2(1) + 1 = 2 + 1 = 3
(1,3)
2
y = 2|2| + 1 = 2(2) + 1 = 4 + 1 = 5
(2,5)
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 1.3
Graphing an Equation
p 29-31
CONTINUED
(-2,5)
 (2,5)
(-1,3)

(1,3)

In graphing an equation, we
try to get enough ordered pairs
to get a good idea of what the
graph looks like. Next,
we plot these points. Finally, we
connect these points with a
smooth curve or line, always
moving from left to right. This
often gives us a picture of all
ordered pairs that satisfy the
equation.
(0,1)
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 1.3
Graphing an Equation
p 29-31
EXAMPLE
Graphs in the rectangular coordinate system can also be used to tell a
story. Try to select the graph that best illustrates the story of the
population of the U.S.A.
Years
Population
(c)
Population
(b)
Population
(a)
Years
Years
SOLUTION
Graph (c)
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 1.3
Graphing an Equation
Do Check 4 on page 31
Look at Example 5 on page 32
Understanding the Viewing Rectangle
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 1.3
p 29-31
§ 2.2
Graphs of Functions
Graphs of Functions
105
The graph of a function is just the
graph of its ordered pairs.
For example, the graph of y = 3x is the set of
points (x, y) satisfying y = 3x.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.2
Graphs of Functions
106
EXAMPLE
Graph the function f x   3x  1 .
SOLUTION
x
y = -3x+1
Ordered Pair (x,y)
-2
f(x) = -3(-2) + 1 = 6 + 1 = 7
(-2,7)
-1
f(x) = -3(-1) + 1 = 3 + 1 = 4
(-1,4)
0
f(x) = -3(0) + 1 = 0 + 1 = 1
(0,1)
1
f(x) = -3(1) + 1 = -3 + 1 = -2
(1,-2)
2
f(x) = -3(2) + 1 = -6 + 1 = -5
(2,-5)
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.2
Graphs of Functions
CONTINUED





Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.2
The Vertical Line Test for Functions
If any vertical line intersects a graph in more than one point,
the graph does not define y as a function of x.
(a)
(b)
(c)
Use the vertical line test to identify graphs in which y is a
function of x.
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.2
107
The Vertical Line Test
107
CONTINUED
SOLUTION
Here, three values of y correspond to one value of x. Whatever the
exact values are, in this graph it is clear that y is not a function of x. The
vertical line test is just a quick visual method for determining whether
you have a function.
(a)
y is a function of x
(b)
y is not a function of x
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.2
(c)
y is a function of x
Graphs of Functions
107
EXAMPLE
The figure shows the cost of mailing a first-class letter, f(x),
as a function of its weight, x, in ounces. Use the graph to
answer the following questions.
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.2
Graphs of Functions
CONTINUED
(a) Find f (3). What does this mean in terms of the variables in
this situation?
(b) Find f (4). What does this mean in terms of the variables in
this situation?
(c) What is the cost of mailing a letter that weighs 1.5 ounces?
(d) What is the cost of mailing a letter that weighs 1.8 ounces?
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.2
Graphs of Functions
CONTINUED
SOLUTION
(a) Find f (3). What does this mean in terms of the variables in
this situation?
f (3) = 0.83. This means that
when a first-class letter weighs 3
ounces, postage costs 83 cents.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.2
Graphs of Functions
CONTINUED
(b) Find f (4). What does this mean in terms of the variables in
this situation?
f (4) = 1.06. This means that
when a first-class letter weighs 4
ounces, postage costs $1.06.
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 2.2
Graphs of Functions
CONTINUED
(c) What is the cost of mailing a letter that weighs 1.5 ounces?
f (1.5) = 0.60. This means that
when a first-class letter weighs
1.5 ounces, postage costs $0.60.
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 2.2
Graphs of Functions
CONTINUED
(d) What is the cost of mailing a letter that weighs 1.8 ounces?
f (1.8) = 0.60. This means that
when a first-class letter weighs
1.8 ounces, postage costs $0.60.
Blitzer, Intermediate Algebra, 5e – Slide #18 Section 2.2
Graphs of Functions
bottom of 107
Obtaining Information from Graphs
A closed dot indicates that the graph does not extend beyond this point and the
point belongs to the graph.
An open dot indicates that the graph does not extend beyond this point and the
point does not belong to the graph.
An arrow indicates that the graph extends indefinitely in the direction in which
the arrow points.
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 2.2
Graphs of Functions
Problem 2 on page 111
(similar to 1 and 3 in homework)
graph the given functions, f and g, in the
same rectangular coordinate system.
Select integers for x, starting with -2 and
ending with 2. Describe how the graph of
g is related to the graph of f.
Blitzer, Intermediate Algebra, 5e – Slide #20 Section 2.2
106, 111
f ( x)  x and g ( x)  x  4
Graphs of Functions
Problem 4 on page 112
(similar to 1 and 3 in homework)
106, 112
f ( x)  2 x and g ( x)  2 x  3
graph the given functions, f and g, in the
same rectangular coordinate system.
Select integers for x, starting with -2
and ending with 2. Describe how the
graph of g is related to the graph of f.
Blitzer, Intermediate Algebra, 5e – Slide #21 Section 2.2
Graphs of Functions
Problem 6 on page 112
(similar to 5 in homework)
graph the given functions, f and g, in the
same rectangular coordinate system.
Select integers for x, starting with -2
and ending with 2. Describe how the
graph of g is related to the graph of f.
Blitzer, Intermediate Algebra, 5e – Slide #22 Section 2.2
106, 112
f ( x)  x 2 and g ( x)  x 2  2
Graphs of Functions
Problem 8 on page 112
(similar to 7 in homework)
graph the given functions, f and g, in the
same rectangular coordinate system.
Select integers for x, starting with -2
and ending with 2. Describe how the
graph of g is related to the graph of f.
Blitzer, Intermediate Algebra, 5e – Slide #23 Section 2.2
106, 112
f ( x)  x and g ( x)  x 1
Graphs of Functions
Problem 10 on page 112
(similar to 9 in homework)
Blitzer, Intermediate Algebra, 5e – Slide #24 Section 2.2
106, 112
f ( x)  x3 and g ( x)  x3 1
Domain and Range
110-111
The graph of a function can be used to
determine the function’s domain and range.
Domain: set of inputs (found on the x axis – the collection of
all x values in the graph)
Range: set of outputs (found on the y axis – the collection of
all y values in the graph)
Blitzer, Intermediate Algebra, 5e – Slide #25 Section 2.2
Domain and Range
EXAMPLE
(similar to 31, 35, and 37 in homework)
Do Check Point 4 on page 111 (picture in the textbook)
a.
domain: x | 2  x  1
b.
domain: x | 2  x  1
c.
domain: x | 3  x  0
range: y | 0  y  3
range: y | 1  y  2
range: y | y  3,2,1
Blitzer, Intermediate Algebra, 5e – Slide #26 Section 2.2
Domain and Range
Problem 42 on page 113
Use the graph of f to determine each of the following
a. dom ainof f
x | x  6
b. range of f
y | y  1
c. f (4)
1
Blitzer, Intermediate Algebra, 5e – Slide #27 Section 2.2
DONE