nm standards: afg.d.5, afg.d.6. slope of a line the ratio between a line’s vertical rise to it...
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NM Standards: AFG.D.5, AFG.D.6
Slope of a LineSlope of a Line
run horizontal
rise verticalslope
The ratio between a line’s vertical rise to it horizontal run.
(Ratio – a comparison of two numbers, usually written as a fraction)
Slope FormulaSlope Formula
12
12
x
y
x
yslope
From (–3, 7) to (–1, –1), go down 8 units and right 2 units.
Example 3-1aExample 3-1a
),( 11 yx
),( 22 yx
Find the slope of the line.
Answer: – 4
Rise/Run Method is not as accurate as the formula in some instances.
Use the formula:
42
8
31
71
12
12
xx
yym
Use the slope formula.
Answer: undefined
Find the slope of the line.
Let be and be .
),( 11 yx
),( 22 yx
Find the slope of the line.
Answer:
),( 11 yx
),( 22 yx
Find the slope of the line.
Answer: 0
),( 11 yx ),( 22 yx
d. Find the slope of the line.
Answer: Undefined
Rate of ChangeRate of Change
Another name for slope which describes how a quantity is changing over time.
RECREATION For one manufacturer of camping equipment, between 1990 and 2000, annual sales increased by $7.4 million per year. In 2000, the total sales were $85.9 million. If sales increase at the same rate, what will be the total sales in 2010?
Slope formula
And let (x2, y2) = (85.9, y2)
Multiply each side by 10.
Add 85.9 to each side.
Simplify.
The coordinates of the point representing the sales for 2010 are (2010, 159.9).
Answer: The total sales in 2010 will be about $159.9 million.
CELLULAR TELEPHONES Between 1994 and 2000, the number of cellular telephone subscribers increased by an average rate of 14.2 million. In 2000, the total subscribers were 109.5 million. If the number of subscribers increase at the same rate, how many subscribers will there be in 2005?
Answer: about 180.5 million
Slope of Parallel & Perpendicular Slope of Parallel & Perpendicular LinesLines
other. each of sreciprocal and
sign opposite are slopes thebecause
slope.exact same thehave thebecause ||
nm
ml
Postulates (Rules)Postulates (Rules)
Two non vertical lines have the same slope if and only if they are parallel.
If and only if means the inverse of the postulate is also true, so:
If two non vertical lines are parallel, then they have the same slope.
Two non vertical lines are perpendicular if and only if the product of their slopes is -1 (or opposite reciprocals of each other)
Write the inverse of the postulate:
The product of the slope of two lines is -1 if and only if the lines are perpendicular.
The slopes are not the same,
The product of the slopes is
are neither parallel nor
perpendicular.
Answer:
Example 3-3aExample 3-3a
Determine whether and are parallel, perpendicular, or neither.
Answer: The slopes are the same, so are parallel.
Determine whether and are parallel, perpendicular, or neither.
Graph the line that contains Q(5, 1) and is parallel to with M(–2, 4) and N(2, 1).
Substitution
Simplify.
Slope formula
The slopes of two parallel lines are the same.
Graph the line. Answer:
The slope of the line parallel to
Start at (5, 1). Move up 3 units and then move left 4 units.
Label the point R.
HW #1: Page 142HW #1: Page 142