![Page 1: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/1.jpg)
P.2
coordinates, lines and increment
![Page 2: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/2.jpg)
Rectangular Coordinate System
The horizontal line is called the x-axis.
The vertical line is called the y-axis.
The point of intersection is the origin.
x-axis
y-axis
origin
Quadrant IQuadrant II
Quadrant III
Quadrant IV
![Page 3: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/3.jpg)
The four regions in the x-y plane are known as quadrants, labeled as follows:
x
yQuadrant I
x > 0, y > 0
Quadrant IV
x > 0, y < 0
Quadrant III
x < 0, y < 0
Quadrant II
x < 0, y > 0
![Page 4: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/4.jpg)
Example : Plotting Points
Plot the point (3,2). Start at the origin and move 3 units to the right. From that point,move 2 units up. Now plot your point
![Page 5: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/5.jpg)
Plotting Points Each point in the xy-
plane corresponds to a unique rdered pair (a, b).
Plot the point (2, 4). Move 2 units right Move 4 units up
2 units
4 units
![Page 6: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/6.jpg)
Increment and distance
When a particle moves from one points in the plane to another, the net changes in its coordinates are called increments.
![Page 7: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/7.jpg)
1 2x x x
An increment in a variable is a net change in the that variable. If x changes from x1 to x2, the increment in x is
Definition
![Page 8: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/8.jpg)
Theorem: Distance FormulaTheorem: Distance Formula
The distance between two points
P x y1 1 1 , and P x y
2 2 2 , , denoted
by d P P1 2, is
2 2 2 2
1 2 2 1 2 1,d P P x y x x y y
![Page 9: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/9.jpg)
Example: Find the distance between the points (3,8) and (-1,2)
P P1 23 8 1 2 , , ,
d P P x x y y1 2 2 1
2
2 1
2,
d P P1 2
2 21 3 2 8,
d P P1 2
2 24 6,
d P P1 2 16 36, 52 2 13
![Page 10: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/10.jpg)
Midpoint Formula The midpoint of a line segment
with endpoints (x1, y1) and (x2, y2) is
1 2 2 1, .2 2
x x y y
![Page 11: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/11.jpg)
Midpoint Formula
Find the midpoint M of the segment with endpoints (10, 5) and (6, 4).
10 ( 6) ( 5 4) 1, 2,
2 2 2
![Page 12: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/12.jpg)
Finding Ordered Pairs that are Solutions of Equations
For the following equation find three ordered pairs that are solutions Of the equation y = 5x+2 Let y = 3
3 = 5x + 25 = 5x1 = x (1,3)
Let x = 0y = 5(0) + 2y = 2(0, 2)
Let x = 1y = 5(1) + 2y = 7(1, 7)
![Page 13: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/13.jpg)
Definitions: X-Intercept The x-intercept is a point on any
graph where the graph touches the x-axis.
The y-coordinate of the x-intercept is always zero.
The x-intercept is denoted by the point (x ,0), where x is any real number.
The x-intercept is also known as the zero or root of an equation.
![Page 14: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/14.jpg)
Definitions: Y-Intercept The y-intercept is a point on
any graph where the graph touches the y-axis.
The x-coordinate of the y-intercept is always zero.
The y-intercept is denoted by the point (0, y), where y is any real number.
![Page 15: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/15.jpg)
Procedure for Finding Intercepts
1. To find the x-intercept(s), if any, of the graph of an equation, let y = 0 in the equation and solve for x.
2. To find the y-intercept(s), if any, of the graph of an equation, let x = 0 in the equation and solve for y.
![Page 16: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/16.jpg)
Example: Find the x- and y-intercepts of the graph of y = x2 + 4x – 5.
To find the x-intercepts, let y = 0 and solve for x.0 = x2 + 4x – 5 Substitute 0 for y.
0 = (x – 1)(x + 5) Factor.
x – 1 = 0 x + 5 = 0
x = 1 x = –5 Solve for x.
So, the x-intercepts are (1, 0) and (–5, 0).
![Page 17: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/17.jpg)
Example: Find the x- and y-intercepts of the graph of y = x2 + 4x – 5.
To find the y-intercept, let x=0 and solve for y.
y = 02 + 4(0) – 5 = –5
So, the y-intercept is (0, –5).
![Page 18: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/18.jpg)
Graph of the Functions
![Page 19: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/19.jpg)
The graph of an equation in two variables x and y consists of the set of points in the xy-plane whose coordinates (x,y) satisfy the equation.
![Page 20: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/20.jpg)
Graphing The graph of an equation is found
by plotting points that are solutions of the equation.
The intercepts of the graph are good points to find first.
x-intercept is an x-value where the graph intersects the x-axis. y = 0
y-intercept is a y-value where the graph intersects the y-axis. x = 0
![Page 21: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/21.jpg)
Graphing an Equation by Point Plotting Step 1 Find the intercepts. Step 2 Find as many additional
ordered pairs as needed. Step 3 Plot the ordered pairs
from Steps 1 and 2. Step 4 Connect the points from
Step 3 in a smooth line or curve.
![Page 22: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/22.jpg)
Example: Sketch the graph of y = –2x + 3.
x y (x, y)
–2 7 (–2, 7)
–1 5 (–1, 5)
0 3 (0, 3)
1 1 (1, 1)
3/2 0 (3/2, 0)
Step 1: Find the intercepts.Step 2: Find as many additional ordered pairs as needed.
![Page 23: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/23.jpg)
Example: Sketch the graph of y = –2x + 3.
3. Plot the points in the coordinate plane.
4 8
4
8
4
–4
x
y
x y (x, y)
–2 7 (–2, 7)
–1 5 (–1, 5)
0 3 (0, 3)
1 1 (1, 1)
3/2 0 (3/2, 0)
![Page 24: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/24.jpg)
4. Connect the points with a straight line.
4 8
4
8
4
–4
x
y
Example: Sketch the graph of y = –2x + 3.
![Page 25: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/25.jpg)
Example 2: Graphing Intercepts
Graph 4y + 5x = 20.Substitute zero for x: 4y = 20 or y =
5.Hence, the y-intercept is (0,5).Substitute zero for the y: 5x = 20 or
x = 4.Hence, the x-intercept is (4,0).
![Page 26: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/26.jpg)
![Page 27: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/27.jpg)
Example
Graph the equation y = 5x + 2
31
0-2/5
20
yx
![Page 28: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/28.jpg)
Example: Sketch the graph of y = x 2.
x y (x, y)
–2 4 (–2, 4)
–1 1 (–1, 1)
0 0 (0, 0)
1 1 (1, 1)
2 4 (2, 4)
3 9 (3, 9)
4 16 (4, 16)
y
x2 4
2
6
8
–2
![Page 29: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/29.jpg)
Example: Sketch the graph of y = | x | .
x y (x, y)
–2 2 (–2, 2)
–1 1 (–1, 1)
0 0 (0, 0)
1 1 (1, 2)
2 2 (2, 2)
y
x–2 2
2
4
![Page 30: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/30.jpg)
Example: Is (3,5) on the graph of ?42 xy
Substitute x = 3 and y = 5 into the equation:
?435 2
495 True!
Therefore, (3,5) is on the graph of the equation.
![Page 31: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/31.jpg)
Definition:
The standard form of an equation of a circle with radius r and center (h, k) is
x h y k r 2 2 2
![Page 32: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/32.jpg)
Definition: A circle is a set of points in the xy-plane that are a fixed distance r from a fixed point (h, k). The fixed distance r is called the radius, and the fixed point (h, k) is called the center of the circle.
x
y
(h, k)
r(x, y)
![Page 33: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/33.jpg)
Standard form of an equation of a circle
where the center of the circle is at the origin (0,0) and with a radius of r.
2 2 2rx y
![Page 34: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/34.jpg)
Example Graph x2 + y2 = 16
3
3
04
40
yx
7
7
![Page 35: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/35.jpg)
Unit Circle equation
where the center of the circle is atthe origin (0,0) and with a radius of1, is called the unit circle.
2 2 1x y
Since the radius = 1, use the center (0,0) as a reference point and then move 1 point to the left, right, up and down.
![Page 36: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/36.jpg)
![Page 37: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/37.jpg)
Continued.
![Page 38: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/38.jpg)
Continued.
![Page 39: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/39.jpg)
Equation of Straight lines
![Page 40: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/40.jpg)
Straight linesDefinition:
1 1 2 2
2 1
2 1
2 1
The of the line through the
distinct points ( , ) and ( , )
Change in Riseis
Change in Run
where
o e
0
sl p
.
x y x y
y y y y
x x x x
x x
![Page 41: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/41.jpg)
2 1
2 1
yThe Slope of the line = tan = .
x
yy
x x
![Page 42: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/42.jpg)
Possibilities for a Line’s Slope
Positive Slope
0m
Line rises from left to right.
![Page 43: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/43.jpg)
Possibilities for a Line’s Slope
Negative Slope
0m
Line falls from left to right.
![Page 44: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/44.jpg)
Possibilities for a Line’s Slope
Zero Slope
0m
Line is horizontal.
![Page 45: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/45.jpg)
Possibilities for a Line’s Slope
Undefined Slope is
undefined.
m
Line is vertical.
![Page 46: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/46.jpg)
Example: Find the slope of the line passing
through the pair of points (2,1) and (3, 4).
1 1 2 2Let ( , ) (2,1) and
Solu
( , ) (3,4).
tion
x y x y
2 1
2 1
Slopey y
mx x
4 1
3 2
3
1 3.
![Page 47: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/47.jpg)
Example: Find the Slope
Find the slope of the line passing through
the pair of points ( 1,3) and (2,4) or state
that the slope is undefined.
![Page 48: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/48.jpg)
Solution
1 1 2 2Let ( , ) ( 1,3) and ( , ) (2,4).x y x y
2 1
2 1
Slopey y
mx x
4 3
2 ( 1)
1
3
The slope is and
the line from l
positi
eft to
ve,
r riises ght.
![Page 49: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/49.jpg)
Practice Exercise
Find the slope of the line passing through
the points (4, 1) and (3, 1) or state
that the slope is undefined.
![Page 50: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/50.jpg)
Answer
The slope is zero.
Thus, the line is a horizontal line.
m
![Page 51: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/51.jpg)
Equation of a Horizontal Line
A horizontal line
is given by an
equation of the
form
where is the
-intercept.
b
b
y
y
Y-interceptis 40m
The graph of 4y
![Page 52: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/52.jpg)
Equation of a Vertical Line
A vertical line is
given by an
equation of the
form
where is the
-intercept.
a
x
x a
X-intercept is -5
Slope is
undefined
The graph of -5x
![Page 53: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/53.jpg)
Example: Draw the graph of the equation x = 2.
y
x
x = 2
![Page 54: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/54.jpg)
Example : Graphing a Horizontal Line
Graph 5 in the
rectangular coordinate system.
y
Y-intercept is 5.
![Page 55: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/55.jpg)
Practice Exercises
Graph each equation in the rectangular
coordinate system.
1. 4
2. 0
y
x
![Page 56: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/56.jpg)
Answers to Practice Exercises
2..1
![Page 57: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/57.jpg)
The Equation of the line
1 2 1
1 2 1
.y y y y
mx x x x
1- With two points P1 (x1 , y1 ) and P2 (x2 , y2 )
1 1 1 2 2 2
The equation of the line passing throuhg the points
P (x , y ) and P (x , y ) is
![Page 58: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/58.jpg)
Example
Write the point-slope of the equation
of the line passing throuhg the points
(3,5) and (8,15). Then solve the
equation for y.
![Page 59: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/59.jpg)
Solution
1 2 1
1 2 1
5 15 5 102
3 8 3 5
y y y y
x x x x
y
x
( 5) 2( 3)y x Then solve for gives:
2 1
y
y x
![Page 60: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/60.jpg)
The Equation of the line
1 1
The point-slope equation of a nonvertical
line of slope that passes through the
point ( , ) is
m
x y
1 1( .)y y m x x
2- Point-slope Form of the Equation of a Line
![Page 61: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/61.jpg)
Example : Writing the Point-Slope Equation of a Line
1 1
We use the point-slope equation of a line
with
Solut
4, 1, and 3.
ion
m x y
Write the point-slope form of the equation
of the line passing through (1,3) with a slope
of 4. Then solve the equation for .y
![Page 62: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/62.jpg)
1 14, 1, and 3m x y
1 1( )y y m x x 3 4( 1)y x 3 4 4y x
4 1y x
![Page 63: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/63.jpg)
Practice Exercises
1. Write the point-slope form of the equation
of the line passing through (4,-1) with a slope
of 8. Then solve the equation for .y
2. Write the point-slope form of the equation
of the line passing through the points ( 2,0)
and (0,2). Then solve the equation for .y
![Page 64: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/64.jpg)
Answers to Practice Exercises
1. 8 33
2. 2
y x
y x
![Page 65: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/65.jpg)
3. The Slope-Intercept Form of the Equation of a Line
The slope-intercept
equation of a
nonvertical line
with slope and
-intercept is
m
y b
y mx b
(0, )b
y
x
Y-intercept is b
Slope is m
A line with slope
and -intercept .
m
y b
![Page 66: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/66.jpg)
Example : Graphing by Using the Slope and y-Intercept
Give the slope and the -intercept of the
line 3 2. Then graph the line.
y
y x
Solution 3 2y x
The slope
is 3
The -intercept
is 2.
y
![Page 67: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/67.jpg)
The graph of 3 2.y x
First use the -intercept 2, to
plot the point (0,2). Starting
at (0,2), move 3 units up and
1 unit to the right. This gives
us the second point of the line.
Use a straightedge to draw a
line through the tw
y
o points.
![Page 68: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/68.jpg)
Text Example
Graph the line whose equation is y = 2/3 x + 2.
Solution:
y = 2/3 x + 2 The slope is
2/3.
The y-intercept is 2.
![Page 69: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/69.jpg)
Text Example cont.Graph the line whose equation is y = 2/3x + 2.
We plot the second point on the line by starting at (0, 2), the first point. Then move 2 units up (the rise) and 3 units to the right (the run). This gives us a second point at (3, 4).
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
![Page 70: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/70.jpg)
Practice Exercises
Give the slope and -intercept
of each line whose equation is
given. Then graph the line.
y
1. 3 2
32. 3
4
y x
y x
![Page 71: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/71.jpg)
Answers to Practice Exercises
1. 3, 2m b 32. , 3
4m b
![Page 72: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/72.jpg)
General Form of the Equation of a Line
0
Every line has an equation that can
be written in the general form
where, , , and are three
real numbers, and and
are not both zero.
A B C
A B
Ax By C
![Page 73: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/73.jpg)
Equations of Lines
1 2 1
1 2 1
1 1
1. Two points :
2. Point-slope form:
3. Slope-intercept form:
4. Horizontal line:
5. r
(
e
)
V t
y y y y
x x x x
y y m x x
y mx b
y b
ical line:
6. General form: 0
x a
Ax By C
![Page 74: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/74.jpg)
Example : Finding the Slope and the y-Intercept
Find the slope and the -intercept of the
line whose equation is 4 6 12 0.
y
x y
SolutionFirst rewrite the equation in slope-intercept
form . We need to solve for .y mx b y
![Page 75: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/75.jpg)
4 6 12 0x y
6 4 12y x 4 12
6 6y x
22
3y x
23
The coefficient of ,
, is the slope and
the constant term, 2,
is the -intercept.
x
y
23 , 2.m b
![Page 76: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/76.jpg)
Practice Exercises
a. Rewrite the given equation in
slope-intercept form.
b. Give the slope and y-intercept.
c. Graph the equation.
1. 6 5 20 0
2. 4 28 0
x y
y
![Page 77: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/77.jpg)
Answers to Practice Exercises
651. 4
6slope
5-intercept 4.
y x
m
y b
![Page 78: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/78.jpg)
Answers to Practice Exercises
2. 7
slope 0
-intercept 7.
y
m
y b
![Page 79: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/79.jpg)
Definitions: Parallel Lines
Two lines are said to be parallel if they do not have any points in common.
Two distinct non-vertical lines are parallel if and only if they have the same slope and have different y-intercepts.
![Page 80: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/80.jpg)
Definitions: Perpendicular Lines
Two lines are said to be perpendicular if they intersect at a right angle.
Two non-vertical lines are perpendicular if and only if the product of their slopes is -1.
121 mm1
2
1
mm
![Page 81: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/81.jpg)
Parallel and PerpendicularLines
Example the following lines are perpendicular
53
123
xyandxy
![Page 82: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/82.jpg)
Text Example
Write an equation of the line passing through (-3, 2) and parallel to the line whose equation is y = 2x + 1. Express the equation in point-slope form and y-intercept form.
![Page 83: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/83.jpg)
Text Example cont.
y – y1 = m(x – x1)
y1 = 2 x1 = -3
Solution . Notice that the line passes through the point (-3, 2). Using the point-slope form of the line’s equation, we have x1 = -3 and y1 = 2.
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
(-3, 2)
Rise = 2
Run = 1
y = 2x + 1
![Page 84: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/84.jpg)
Text Example cont.
Solution Parallel lines have the same slope. Because the slope of the given line is 2, m = 2 for the new equation.
-5 -4 -3 -2 -1 1 2 3 4 5
5
4
3
2
1
-1
-2
-3
-4
-5
(-3, 2)
Rise = 2
Run = 1
y = 2x + 1
y – y1 = m(x – x1)
y1 = 2 m = 2 x1 = -3
![Page 85: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/85.jpg)
Text Example cont.Solution The point-slope form of the line’s equation is y – 2 = 2[x – (-3)]
y – 2 = 2(x + 3)
Solving for y, we obtain the slope-intercept form of the equation.
y – 2 = 2x + 6
y = 2x + 8
![Page 86: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/86.jpg)
Text Example
Write an equation of the line passing through (-3, 2) and perpendicular to the line whose equation is y = 2x + 1. Express the equation in point-slope form and y-intercept form.
![Page 87: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/87.jpg)
Text Example cont.
y – y1 = m(x – x1)
y1 = 2 x1 = -3
Solution Using the point-slope form of the line’s equation, we have x1 = -3 and y1 = 2.
![Page 88: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/88.jpg)
Text Example cont.
Solution perpendicular lines have the product of their slopes is -1. Because the slope of the given line is 2, m = -1/2 for the new equation.
y – y1 = m(x – x1)
y1 = 2 m = -1/2 x1 = -3
![Page 89: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/89.jpg)
Text Example cont.Solution The point-slope form of the line’s equation is y – 2 = -1/2[x – (-3)]
y – 2 = -1/2(x + 3)
Solving for y, we obtain the slope-intercept form of the equation.
y – 2 = -1/2x -3/2
y = -1/2x + 1/2
![Page 90: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/90.jpg)
Find the equation of the line parallel to y = -3x + 5 passing through (1,5).
Since parallel lines have the same slope, the slope of the parallel line is m = -3.
y y m x x 1 1
y x 5 3 1
y x 5 3 3y x 3 8
![Page 91: P.2 coordinates, lines and increment. Rectangular Coordinate System The horizontal line is called the x-axis. The vertical line is called the y-axis](https://reader035.vdocument.in/reader035/viewer/2022062718/56649e6b5503460f94b6980b/html5/thumbnails/91.jpg)
Example: Find the equation of the line perpendicular to y = -3x + 5 passing through (1,5).
Slope of perpendicular line:
13
13
y y m x x 1 1
y x 513
1
y x 513
13
y x 13
143