Linear Functions
Lesson 1: Slope of a Line
Today’s Objectives
• Demonstrate an understanding of slope with
respect to: rise and run; rate of change; and line
segments and lines, including:
• Determine the slope of a line or line segment using
rise and run
• Classify a line as having either positive or negative
slope
• Explain the slope of a horizontal or vertical line
• Explain why the slope can be found using any two
points on the graph of the line or line
• Draw a line segment given its slope and a point on the
line
Vocabulary
• Slope
• The measure of a lines steepness
• (vertical change/horizontal change)
• Rise
• The vertical change of a line
• Run
• The horizontal change of a line
Slope of a Line
• The slope of a line segment is a measure of its steepness
• This means a comparison between the vertical change
and the horizontal change:
• The vertical change (∆𝑦)is called the rise
• The horizontal change (∆𝑥) is called the run
• Slope is normally represented by the lowercase m.
• We can calculate the slope in several ways such as by
counting or using coordinates of two points on the line
• m = slope =
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
=
∆𝑦
∆𝑥
=
𝑦2−𝑦1
𝑥2−𝑥1
Slope of a Line
A) Counting
B) Slope formula
Slope = rise/run
Slope = -3/6
Slope = -1/2
(x1,y1)
A (-2,1)
Slope = rise/run = y2-y1/x2-x1
Slope = [-2-1]/[4-(-2)]
Slope = -3/6 = -1/2
Down 3
(x2,y2)
B(4,-2)
Right 6
Slope of a Line
• If the line segment goes downward from left to
right, it will have a negative slope. (rise =
negative)
• If the line segment goes upwards from left to
right, it will have a positive slope. (rise =
positive)
• *The steeper the line goes up or down, the
greater the slope.
Horizontal and Vertical Lines
• If a line is horizontal, that is, the rise is equal to
zero, then the slope will also be equal to zero.
•
Slope
rise
0
= =
run run
=0
• If a line is vertical, that is, the run is equal to
zero, then the slope of the line will be
undefined.
•
Slope =
rise
= =
run
rise
=
0
= undefined
Example 1) You do
• Find the slopes of the following line segments.
Which line segment has the steepest (greatest)
slope? Graph the line segments.
• A) A(-1, 7)
B(4, -3)
• B) A(-20, 3)
B(-4, -5)
Solutions
(-1,7)
(-20,3)
(4,-3)
(-4,-5)
Slope of line a) = -10/5 = -2
Slope of line b) = -8/16 = -1/2
Line segment in a) is steeper than line segment b)
Finding Unknown Coordinates
• We can also use the slope formula to find the coordinates of an
unknown point on the line when we know the slope and
another point on the line.
• Example 2) Given a line that passes through R(5,-6) and has a
slope of -2/7, determine another point, T, that the line passes
through.
• Solution:
• We can set one unknown coordinate to equal zero, then solve
for the final remaining unknown coordinate. For example:
• Let x = 0, solve for y
•
−2 y− −6 y+6
=
=
7
0−5
−5
;y+6=
10
7
• y = 10/7 – 6 = -32/7. So a second point, T, on the graph could be
(0, -32/7)
Finding Unknown Coordinates
• Another way to find a second point is to simply count out the
rise and the run from the one known point.
• In this case we can read the slope as -2/7 or 2/-7, so we could
find two possible points: (-2, -4) or (12, -8)
(-2, -4)
Known point (5, -6)
(12, -8)
Wall Quiz!
Homework
• Pg. 339-343
• #4,6,9,10,16,17,20,22,24,26,29