3-5
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the slope of a line is the number that describes the steepness of the line  any ___ points
on a line can be used to determine the slope
Definitions:
rise: the difference in the y- values of two
points on a line
run: the difference in the x-values of two
points on a line
slope: of a line is the ratio of rise to run.
Examples:
If (x1, y1) and (x2, y2) are any two points on a
line, then the slope of the line is
y2 - y1
m = ----------x2 - x1
Ex.1: Use the slope formula to determine the slope of each line.
A. line AB
first identify the points:
A: (___, ____)  (x1, y1)
B: (___, ____)  (x2, y2)
y2 - y1
m = ----------- = --------- = ----- =
x2 - x1
B. line AC
first identify the points:
A: (___, ____)  (x1, y1)
C: (___, ____)  (x2, y2)
y2 - y1
m = ----------- = --------- = ----- =
x2 - x1
C. line AD
first identify the points:
A: (___, ____)  (x1, y1)
D: (___, ____)  (x2, y2)
y2 - y1
m = ----------- = --------- = ----- =
x2 - x1
D. line CD
first identify the points:
C: (___, ____)  (x1, y1)
D: (___, ____)  (x2, y2)
y2 - y1
m = ----------- = --------- = ----- =
x2 - x1
What does the slope of a line look like???
Positive Slope
Negative Slope
Zero Slope
Undefined Slope
-- sometimes slope is used when we talk about miles per hour – as in the rate of change – then
x represents time in hours
and y represents miles traveled
{say for instance….in our next example!}
Ex. 2: Stephen is driving from home to his college dormitory. At 4:00 P.M., he is 260 miles
from home. At 7:00 P.M., he is 455 miles from home. Graph the line that represents
Stephen’s distance from home at a given time. Find and interpret the slope of the line.
so our two points follow this set up: (time in hours, miles traveled)
let:
(_____, _____) &
(x1, y1)
(_____, ______)
(x2, y2)
y2 - y1
m = ----------- = --------------- = -------- =
x2 - x1
Stephen is traveling at an average speed of _______
Theorems:
Parallel Lines Theorem
Perpendicular Lines Theorem
In a coordinate plane, two nonvertical lines are
parallel if and only if they have the same slope.
Any two vertical lines are parallel.
In a coordinate plane, two nonvertical lines are
perpendicular if and only if the product of their
slopes is -1 {this means they are opposite
reciprocals.} Vertical and horizontal lines are
perpendicular.
-- what do these mean???
a
b
 if a line has a slope of ----- , then the slope of a perpendicular line is __ ---b
a
 these ratios are called opposite reciprocals
Ex. 3: Graph each pair of lines. Use their slopes to determine whether they are parallel,
perpendicular or neither.
A. line UV and line XY where U(0, 2), V(-1, -1), X(3, 1) and Y(-3, 3)
(x1, y1) (x2, y2) (___, ___) (___, ___)
y2 - y1
slope of line UV = ----------- = --------- = ------ =
x2 - x 1
slope of line XY = ----------- = --------- = ------ =
------
Answer: ________________________
______________________________
--------------------------------------------------------------------------------------------------------B. line GH and line IJ where G(-3, -2), H(1, 2), I(-2, 4), and J(2, -4)
(x1, y1) (x2, y2) (___, ___) (___, ___)
slope of line GH = ----------- = --------- = ------ =
slope of line IJ = ----------- = --------- = ------ =
Answer: ________________________
______________________________
--------------------------------------------------------------------------------------------------------C. line CD and line EF where C(-1, -3), D(1, 1), E(-1, 1) and F(0, 3)
(x1, y1) (x2, y2) (___, ___) (___, ___)
slope of line CD = ----------- = --------- = ------ =
slope of line EF = ----------- = --------- = ------ =
Answer: ________________________
______________________________