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MAT 125 – Applied Calculus1.4 Straight Lines
1.4 Straight Lines
2 Today’s Class
We will be learning the following concepts in Section 1.3: The Cartesian Coordinate System
The Distance Formula
The Equation of a Circle
We will be learning the following concepts in Section 1.4: Slope of a Line
Equations of Lines
Dr. Erickson
1.4 Straight Lines
3
Slope of a Nonvertical Line
If (x1, y1) and (x2, y2) are two distinct points on a nonvertical line L, then the slope m of L is given by
2 1
2 1
y yym
x x x
(x1, y1)
(x2, y2)
y
x
L
y2 – y1 = y
x2 – x1 = x
Dr. Erickson
1.4 Straight Lines
4
Slope of a Vertical Line
Let L denote the unique straight line that passes through the two distinct points (x1, y1) and (x2, y2). If x1 = x2, then L is a vertical line, and the slope is undefined.
(x1, y1)
(x2, y2)
y
x
L
Dr. Erickson
1.4 Straight Lines
5
Slope of a Nonvertical Line
If m > 0, the line slants upward from left to right.
y
x
L
y = 2
x = 1
m = 2
Dr. Erickson
1.4 Straight Lines
6
Slope of a Nonvertical Line
If m < 0, the line slants downward from left to right.
m = –1
y
x
L
y = –1
x = 1
Dr. Erickson
1.4 Straight Lines
7
Sketch the straight line that passes through the point (1, 2) and has slope – 2.
1 2 3 4 5 6
y
x
6
5
4
3
2
1
Example 1
Dr. Erickson
1.4 Straight Lines
8
Find the slope m of the line that goes through the points (–2, –2) and (4, –4).
Example 2
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1.4 Straight Lines
9
Let L be a straight line parallel to the y-axis. Then L crosses the x-axis at some point (a, 0), with the x-coordinate given by x = a, where a is a real number. Any other point on L has the form (a, y), where y is an appropriate number. The vertical line L can therefore be described as x = a
Equations of Lines
(a, y )
y
x
L
(a, 0)
Dr. Erickson
1.4 Straight Lines
10 Equations of LinesLet L be a nonvertical line with a slope m.
Let (x1, y1) be a fixed point lying on L and (x, y) be variable point on L distinct from (x1, y1).
Using the slope formula by letting (x2, y2) = (x, y) we get
Multiplying both sides by x – x1 we get
1
1
y ym
x x
1 1( )y y m x x
Dr. Erickson
1.4 Straight Lines
11 Point-Slope Form of an Equation of a Line
An equation of the line that has slope m and passes through point (x1, y1) is given by
Dr. Erickson
1 1( )y y m x x
1.4 Straight Lines
12
Find an equation of the line that passes through the point (2, 4) and has slope –1.
Example 5
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1.4 Straight Lines
13
Find an equation of the line that passes through the points (–1, –2) and (3, –4).
Example 6
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1.4 Straight Lines
14 Parallel Lines
Two distinct lines are parallel if and only if their slopes are equal or their slopes are undefined.
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1.4 Straight Lines
15 Perpendicular Lines
If L1 and L2 are two distinct nonvertical lines that have slopes m1 and m2, respectively, then L1 is perpendicular to L2 (written L1 ┴ L2) if and only if
Dr. Erickson
12
1m
m
1.4 Straight Lines
16
Example 7
Find an equation of the line that passes through the point (2, 4) and is perpendicular to the line
Find an equation of the line that passes through the origin and is parallel to the line joining the points (2,4) and (4,7).
3 4 22 0.x y
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1.4 Straight Lines
17
Crossing the Axis
A straight line L that is neither horizontal nor vertical cuts the x-axis and the y-axis at, say, points (a, 0) and (0, b), respectively.
The numbers a and b are called the x-intercept and y-intercept, respectively, of L.
(a, 0)
(0, b)
y
x
L
y-intercept
x-intercept
Dr. Erickson
1.4 Straight Lines
18 Slope Intercept Form of an Equation of a Line
An equation of the line that has slope m and intersects the y-axis at the point (0, b) is given by
y = mx + b
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1.4 Straight Lines
19 Example 8Find the equation of the line that has the following:
m = –1/2; b = 3/4
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1.4 Straight Lines
20 Example 9Determine the slope and y-intercept of the line whose equation is 3x – 4y + 8=0.
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1.4 Straight Lines
21Example 10
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1.4 Straight Lines
22 General Form of an Linear Equation
The equation
Ax + By + C = 0
where A, B, and C are constants and A and B are not both zero, is called the general form of a linear equation in the variables x and y.
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1.4 Straight Lines
23 Theorem 1
An equation of a straight line is a linear equation; conversely, every linear equation represents a straight line.
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1.4 Straight Lines
24 Example 11Sketch the straight line represented by the equation 3x – 2y +6 = 0.
Dr. Erickson
1.4 Straight Lines
25 Next Class
We will discuss the following concepts: Functions
Determining the Domain of a Function
Graphs of Functions
The Vertical Line Test
Please read through Section 2.1 – Functions and Their Graphs in your text book before next class.
Dr. Erickson
