Download Chapter 1 Section 1.1 – Using Qualitative Graphs to Describe

Chapter 1
Section 1.1 – Using Qualitative Graphs to Describe Situations
Activity #2
1
Section 1.2 – Graphing Linear Equations
We will be looking at linear equations in two variables. Consider the linear equation
3x + 2 y = 6
♦ This is a linear equation because ITS graph is a LINE.
♦ The ordered pair (0,3) is a solution because substituting x=0 and y=3 we get a TRUE
STATEMENT.
♦ Is (1,5) a solution to 3 x + 2 y = 6 ? Why or why Not?
♦ Can you find other solutions?
♦ How many solutions are there?
2
♦ The solution set is the collection of ALL the solutions.
♦ The graph of an equation in two variables is the solution set.
♦ Graph 3 x + 2 y = 6
Example 1
Graph the following equations by plotting 3 points and labeling them on the graph.
a) y = −4 x − 2
3
b) 3 y + 3 x − 2 = 7
c) 5 x − 2(3 y − 1) = −2 y − 3( x − 2 )
4
d) y = −
2
x−2
3
e) Graph BOTH lines on the same grid. x = 4 , y = −2
5
Definitions
♦ X intercept is a point where your graph CROSSES the x axis
♦ Y intercept is a point where your graph CROSSES the y axis
Example #2
Identify the x and y intercepts. If there are none, state why.
y
1.0
y 140
0.8
120
0.6
100
0.4
80
0.2
60
-5
-4
-3
-2
-1
1
2
3
4
-0.2
5
x
40
-0.4
20
-0.6
-0.8
-5
-1.0
y
-4
-3
-2
-1
0
1
2
3
4
5
x
6
4
2
-5
-4
-3
-2
-1
1
-2
2
3
4
5
x
-4
-6
-8
-10
-12
6
Section 1.3 – Slope of a line
• Slope = Steepness = Is a number that measures how steep a line is.
a) Positive slope
b) Negative slope
c) NO slope
d) Undefined slope
7
How to find the slope of a line
e) Formula for the Slope of line
m=
y 2 − y1
x 2 − x1
Example #1
Find the slope of the line passing through the given points. State whether the line is increasing,
decreasing, horizontal or vertical.
a)
(1,8) and (5,4)
⎛2
⎝3
⎞
⎠
⎛ 1 ⎞
,0 ⎟
⎝ 6 ⎠
b) ⎜ ,2 ⎟ and ⎜ −
⎛ 100 ⎞
⎛ − 45 ⎞
,2 ⎟
,2 ⎟ and ⎜
⎝ 234 ⎠
⎝ 117 ⎠
c) ⎜
d)
(1,6) and (1, π )
8
Perpendicular and Parallel Lines
Definitions
a) Slopes of Parallel Lines
y
8
6
4
2
-5
-4
-3
-2
-1
1
2
3
4
3
4
-2
5
x
-4
-6
-8
-10
-12
-14
b) Slopes of perpendicular Lines
y
10
8
6
4
2
-5
-4
-3
-2
-1
1
2
5
x
-2
-4
9
Example #2
State if the following linear equations are parallel, perpendicular or neither.
a) 3 x − 5 y = 6 and y =
b) y =
3
6
x−
5
5
1
x + 5 and 2 y = −4 x + 12
2
c) x = 5 and y = −76
d) 5 x − 6 = 5 y and 3 x −
1
y=6
2
10
Section 1.4 – Meaning of Slope for Equations, Graphs and Tables.
Example #1
Find the slope and y intercept for the following linear equations.
a) y = −
4
x +1
3
b) 3( x − 2 y ) = 9
c) 8 − 2( y − 3 x ) = 2 + 4( x − 2 y )
d) y + 2 = 0
11
e)
x y
+ =1
a b
f)
y+b
=x
a
Example #2
Complete the following table
Table 16 – Values of Four Linear Equations
Equation 1
Equation 2
Equation 3
X
Y
x
y
x
1
12
23
69
2
15
24
53
y
Equation 4
x
Y
1
30
15
2
31
3
25
3
35
4
26
4
33
5
27
5
34
6
28
6
17
32
35
60
12
Example 3
A line contains the points (-6, -4) and (-3,1). Find three more points that lie on the line.
Example 4
Let y be a college’s enrollment (in thousands of students) and let x be the number of years that the
college has been open. Assume that the equation y = 0.5 x + 6 describes the relationship between x and
y.
a) Complete the table below
Number of Years
College has been open
Enrollment ( y )
x
0
1
2
3
4
b) By how much does the college’s enrollment increase each year?
13
Section 1.5 – Finding the equation of a line
Remember that a linear equation is an equation that can be written in the form y = mx + b
where m = SLOPE and (0, b ) is the y-intercept.
To find the equation of a line you need …
♦ Slope
♦ Point on the line
Once you have this information, then we will be looking at two different methods for finding our
equations.
Method 1 – Using the SLOPE -INTERCEPT form
With Method 1, we will be using the SLOPE INTERCEPT FORM. What is the SLOPE INTERCEPT FORM
of any equation? Another way to ask this question is to say …”when is an equation in slope intercept
form?”
So … let’s look at the following examples.
Example #1
a) Find the equation of the line with slope =
y
−1
and passing through the point (− 1,0 ) and GRAPH!!!!
2
5
4
3
2
1
-5
-4
-3
-2
-1
1
-1
2
3
4
5
x
-2
-3
-4
-5
14
a) Find the equation of the line passing through (4,3) and (− 1,4 ) and GRAPH!!
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
-2
-3
-4
-5
⎛1 ⎞
⎝2 ⎠
b) Find the equation of the line with slope = 5 and passing through the point ⎜ ,0 ⎟ and GRAPH!!
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
-1
2
3
4
5
x
-2
-3
-4
-5
15
c) Find the equation of the line with slope=undefined and passing through the point (− 1,5)
d) GRAPH!!
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
-1
5
x
-2
-3
-4
-5
e) Find the equation of the line parallel to 3 x − 2 y + 4 = 0 and passing through the point (4,3)
f) GRAPH both lines!!
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
-1
2
3
4
5
x
-2
-3
-4
-5
16
g) Find the equation of the line PERPENDICULAR to − 3 x − 4 y = 12 and passing through the point
(− 1,2)
h) GRAPH BOTH EQUATIONS
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
-1
4
5
x
-2
-3
-4
-5
METHOD 2 – Point-SLOPE form
In a non-vertical line has slope m and contains the point (x1 , y1 ) , then an equation of the line is
y − y1 = m( x − x1 )
Example 2 (This problem is the same as EXAMPLE #1 ABOVE!!)
Find the equation of the line with slope =
−1
and passing through the point (− 1,0 )
2
17
Example 3
a) Find an equation of the line that contains the point (2,3) and is parallel to y = 6
b) GRAPH!!!
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
-1
5
x
-2
-3
-4
-5
c) Find an equation of the line that contains the point (2,3) and is perpendicular to y = 6
d) GRAPH
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
-1
2
3
4
5
x
-2
-3
-4
-5
18
Example 4 - NUMBER 70 in book
Let y be a person’s salary (in thousands of dollars) after he has worked at a company of x years. Some
pairs of values of x and y are listed below.
TABLE 23 Salaries
Time at
company (years)
x
Salary –
(thousands of
dollars)
y
0
25
1
28
2
31
3
34
4
37
5
40
Find an equation that describes the relationship between x and y .
Example 5
Find an equation of the line sketched below.
14
12
10
8
6
4
2
-5
-4
-3
-2
-1
1
2
3
4
5
-2
-4
-6
19
Section 1.6 – Functions
Definition
A RELATION is a set of ordered pairs. The set of the first coordinates ( x coordinates) is called the
DOMAIN of the relation and the set of the second coordinates ( y coordinates) is called the RANGE of
the relation.
Example of a relation:
Definition
A FUNCTION is a relation where ONE INPUT is mapped to EXACTLY ONE OUTPUT.
Different ways I can describe a function:
a) An equation
b) A table
c) A graph
d) Words (EVALUATION MACHINES) Activity #3
20
Vertical Line Test
A relation is a function IF and ONLY IF every vertical line intersects the graph at NO MORE than
ONE POINT.
Definition
A linear function is a relation whose equation can be put into the form y = mx + b
Finding DOMAINS and RANGES
Activity 3, 4, and 5
21