Transcript Math 025 Section 7.3 Slope

Math 025
Section 7.3
Slope
Objectives:
• to find the x- and y-intercepts of a straight line
• to graph a line using x- and y-intercepts
• to find the slope of a straight line
• to determine from their slopes whether two lines
are parallel
• to graph a line using the slope and the y-intercept
The graph of 2x + 3y = 6 is shown below.
The graph crosses the x-axis at the
point (3, 0)
This point is called the
x-intercept of the graph
y
(0, 2)
(3, 0)
x
The graph crosses the y-axis at the
point (0, 2)
This point is called the
Note:
y-intercept of the graph
y = 0 at the x-intercept
x = 0 at the y-intercept
Find the x-intercept and y-intercept for x – 2y = 4
Then graph the line
x-intercept:
Let y = 0
y
x – 2(0) = 4
x=4
x-intercept: (4, 0)
y-intercept:
Let x = 0
x
0 – 2y = 4
-2y = 4
y = -2
y-intercept: (0, -2)
The slope of a line is a measure of the slant of the line.
It is sometimes described as: Slope = rise
run
It can also be described as: Slope = change in y
change in x
zero slope
undefined
slope
slope = m =
Dy
Dx
If you move in a negative
direction to find Dy or Dx,
use a negative number to
describe that movement.
Red line:
m=
-3
1
Blue line:
m=
5
7
= -3
You can find the slope of a line without graphing it if you
know two points, P1(x1, y1) and P2(x2, y2), that are on the
line.
slope formula:
y2 – y1
m=
x2 – x1
Find the slope of the line containing the given points
P1(1, 3)
P2(5, -3)
m = -3 – 3
5–1
P1(-1, 2)
P2(-1, -3)
m = -3 – 2 = -5
-1 – (-1)
0
= -6
4
= -3
2
= Undefined
Parallel lines have the same slope
Determine whether the line through P1 and P2
is parallel to the line through Q1 and Q2
P1(4, -5)
P2(6, -9)
Q1(5, -4)
Q2(1, 4)
mP = -5 + 9 = 4
-2
4–6
mQ = -4 – 4 = -8
4
5–1
= -2
= -2
The slopes are the same, so the lines are
parallel.
Perpendicular lines have slopes that are negative
reciprocals of each other
Determine whether the line through P1 and P2
is perpendicular to the line through Q1 and Q2
P1(-4, -5)
P2(-6, -9)
Q1(-4, 5)
Q2(4, 1)
mP = -5 + 9 = 4
-4 + 6 2
mQ = 1 – 5 = -4
8
4+4
The slopes are negative reciprocals, so
the lines are perpendicular.
= 2
= -1
2
Determine whether the lines through P1 and P2 and
through Q1 and Q2 are parallel, perpendicular or neither
1+2 = 3
m
=
P
P1(5, 1) P2(3, -2)
2
5–3
Q1(0, -2) Q2(3, -4)
mQ = -4 + 2 = -2
3
3–0
Perpendicular
P1(1, -1)
P2(3, -2)
Q1(-4, 1)
Q2(2, -5)
1
-1
+
2
mP =
=
-2
1–3
mQ = -5 – 1 = -6
6
2+4
Neither
= -1
(-2, 5)
y = -3x + 2
2
Notice that the above
equation can be used to
determine the slope and the
y-intercept without having
seen the graph.
The slope is the
coefficient of x
The y-coordinate of the
y-intercept point is the
constant at the end of the
equation.
(2, -1)
m = 5 + 1 = 6 = -3
-4
2
-2 – 2
y-intercept = (0, 2)
Slope-intercept form of an equation
y = mx + b
Slope = m
y-intercept = (0,b)
Find the slope and the y-intercept of each equation
y = -3x + 5
4
slope = -3
4
y-intercept = (0,5)
2x – 5y = 10
-5y = -2x + 10
y = 2x - 2
5
slope = 2
5
y-intercept = (0, -2)
Graph: y = -2x + 4
5
slope = -2/5
y-intercept = (0, 4)
1st:
Graph the
y-intercept point
2nd:
Use the slope movements
to find another point
3rd:
Draw the line
Graph:
2x – 3y = -3
-3y = -2x – 3
y = 2x + 1
3
slope = 2/3
y-intercept = (0, 1)