1 Equations of the form ax + by = c are called linear equations in two variables. The point (0,4) is...
-
Upload
darrell-griffin -
Category
Documents
-
view
213 -
download
0
Transcript of 1 Equations of the form ax + by = c are called linear equations in two variables. The point (0,4) is...
1
Equations of the form ax + by = c are called linear equations in two variables.
The point (0,4) is the y-intercept.
The point (6,0) is the x-intercept.
x
y
2-2
This is the graph of the equation 2x + 3y = 12.
(0,4)
(6,0)
5.1 Equations of Lines
2
The slope of a line is a number, m, and is defined.
m = 0
m is undefined
negative
riseslope
run
If a line goes up from left to right, then it has a positive slope.
If a line goes down from left to right, then it has a negative slope.
positive
5.1 Equations of Lines
3
y
x2-2
The slope of a line is a number, m, which measures its steepness.
m = 0
m = 2m is undefined
m =1
2
m = -1
4
5.1 Equations of Lines
4
y
x1-1
42
2m
5.1 Equations of Lines
5
x
y
x2 – x1
y2 – y1
change in y
change in x
The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula
The slope is the change in y divided by the change in x as we move along the line from (x1, y1) to (x2, y2).
y2 – y1
x2 – x1
m = , (x1 ≠ x2 ).
(x1, y1)
(x2, y2)
5.1 Equations of Lines
6
Example: Find the slope of the line passing through the points (2, 3) and (4, 5).
Use the slope formula with x1= 2, y1 = 3, x2 = 4, and y2 = 5.
y2 – y1
x2 – x1
m = 5 – 3
4 – 2= =
22
= 1
2
2(2, 3)
(4, 5)
x
y
5.1 Equations of Lines
7
State the slope and the y-intercept y = -2x + 3 x + y = 10 2x = 4 y = 5 x + 2y – 5 = 0 y = x2 + 4
5.1 Equations of Lines
8
In 2000, there were 200 Twins fans at McDonell. The number will grow by 20 each year. Write a linear model and sketch the graph.
5.1 Equations of Lines
N 20t 200
9
5.1 Equations of Lines
N 20t 200
How many fans will there be in 5 years?
300200)5(20 N
When will there be360 fans?360 = 20t + 200
160 = 20t t = 8 years
10
A linear equation written in the form y = mx + b is in slope-intercept form.
To graph an equation in slope-intercept form:
1. Write the equation in the form y = mx + b. Identify m and b.
The slope is m and the y-intercept is (0, b).
2. Plot the y-intercept (0, b).
3. Starting at the y-intercept, find another point on the line using the slope.
4. Draw the line through (0, b) and the point located using the slope.
5.1 Equations of Lines
11
1
Example: Graph the line y = 2x – 4.
2. Plot the y-intercept, (0, - 4).
1. The equation y = 2x – 4 is in the slope-intercept form. So, m = 2 and b = - 4.
3. The slope is 2.
The point (1, -2) is also on the line.
1= change in y
change in xm = 2
4. Start at the point (0, 4). Count 1 unit to the right and 2 units up to locate a second point on the line.
2
x
y
5. Draw the line through (0, 4) and (1, -2).
(0, - 4)
(1, -2)
5.1 Equations of Lines
12
5.2 Equations of LinesPoint Slope
Write an equation of the line that passes through the point (2,-1) with a slope of 2.
A(2,-1)
bmxy b )2(21
b 41
b 5
52 xy
13
5.2 Equations of LinesPoint Slope
Write an equation of the line that passes through the point (-1,-1) with a slope of -3
A(-1,-1)
bmxy b )1(31
b 31
b 4
43 xy
14
5.2 Equations of LinesPoint Slope
Write an equation of the line that passes through the point (1,0) with a slope of 1/2
A(1,0)
bmxy
b )1(2
10
b2
10
b2
1
2
1
2
1 xy
15
5.3 Equations of LinesTwo Points
Write an equation of the line that passes through the point (2,1) and (3,-3)
A(2,1)
12
12
xx
yym
b )2(41
41
4
23
13
m
bmxy
b9
94 xy
B(3,-3)
16
5.3 Equations of LinesTwo Points
Write an equation of the line that passes through the point (1,4) and (0, 3)
A(1,4)
12
12
xx
yym
b )1(14
11
1
01
34
m
bmxy
b3
3xy B(0,3)
17
5.3 Equations of LinesTwo Points
Write an equation of the line that passes through the point (-1,6) and (3, -2)
A(-1,6)12
12
xx
yym
b )1(26
24
8
31
)2(6
m
bmxy
b4
42 xy
B(3,-2)
18
5.3 Equations of LinesTwo Points
Write an equation of the line that passes through the point (1,-3) and (3, -2)
A(-1,-3)
12
12
xx
yym
b )1(2
13
2
1
2
1
31
)2(3
m
bmxy
2
7
2
13 b
2
7
2
1 xy
B(3,-2)
19
5.4 Exploring Data: Fitting a Line to Data
20
5.4 Exploring Data: Fitting a Line to Data
Find the equation of the line using 1928 and 1998
12
12
xx
yym
19201988
2.125.10
25.68
7.1
m
21
5.4 Exploring Data: Fitting a Line to Data
Find the equation of the line using 1928 and 1998
025.68
7.1
m
bmxy
b )1988)(025.(5.10
b 7.495.10
b2.60
2.60025. xy
22
5.4 Exploring Data: Fitting a Line to Data
23
5.4 Exploring Data: Fitting a Line to Data
Correlation: The agreement between two sets of data
-1 < r < 1r is called the correlationcoefficient
24
5.5 Standard Form of a Linear Equation
Form Standard
FormIntercept Slope
cby ax
bmxy
25
5.5 Standard Form of a Linear Equation
Form Standard to12
1Transform xy
12
1 xy Multiply both sides by 2
22 xy Subtract 2y from both sides
220 yx Add 2 to both sides
22 yx
26
5.5 Standard Form of a Linear Equation
Form Standard to4
1
3
2Transform xy
4
1
3
2 xy Multiply both sides by 12
3812 xy Subtract 12y from both sides
31280 yx Add 3 to both sides
3128 yx
27
A linear equation written in the form y – y1 = m(x – x1) is in point-slope form. It is used mainly to write the equation of a line. The graph of this equation is a line with slope m passing through the point (x1, y1).
The graph of the equation
y – 3 = - (x – 4) is a line
of slope m = - passing
through the point (4, 3).
1
2 1
2
(4, 3)
m = -1
2
x
y
4
4
8
8
5.6 Point-Slope Form of a Line
28
Example: Write the slope-intercept form for the equation of the line through the point (-2, 5) with a slope of 3.
Use the point-slope form, y – y1 = m(x – x1), with m = 3 and (x1, y1) = (-2, 5).
y – y1 = m(x – x1) Point-slope form
y – y1 = 3(x – x1) Let m = 3.
y – 5 = 3(x – (-2)) Let (x1, y1) = (-2, 5).
y – 5 = 3(x + 2) Simplify.
y = 3x + 11 Slope-intercept form
5.6 Point-Slope Form of a Line
29
Example: Write the slope-intercept form for the equation of the line through the points (4, 3) and (-2, 5).
y – y1 = m(x – x1) Point-slope form
Slope-intercept formy = - x + 13
31
3
2 1 5 – 3 -2 – 4
= - 6
= - 3
Calculate the slope.m =
Use m = - and the point (4, 3).y – 3 = - (x – 4)1
3 3
1
5.6 Point-Slope Form of a Line
30
Summary of Equations of Lines
5.6 Point-Slope Form of a Line
Slope of a line through two points:
Vertical line (undefined slope):
Horizontal line (zero slope):
Slope-intercept form:
Point-slope form:
Standard form:
12
12
xx
yym
x = a
y = b
y = mx + b
y2 – y1 = m(x2 – x1)
Ax + By = C