c Dr Oksana Shatalov, Fall 2010
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1: STRAIGHT LINES AND LINEAR FUNCTIONS
1.2: Straight Lines
• Slope
DEFINITION 1. If (x1 , y1 ) and (x2 , y2 ) are any two
distinct points on a non-vertical line L, then the slope
m of L is
∆y
y2 − y1
.
m=
=
∆x
x2 − x1
Note that the slope of a straight line is a constant whenever it is defined.
y
x
0
EXAMPLE 2. (a) Plot and label the points A(4, 2), B(2, 1), C(1, 3), D(1, 1), E(0, 3).
(b) Sketch the straight lines L1 , L2 , L3 , L4 , L5 if L1 passes through A and B, L2 passes
through B and C, L3 passes through C and E, L4 passes through C and D, L5 passes
through D and E.
(c) Find the slope of L1 , L2 , L3 , L4 , L5 .
y
x
0
c Dr Oksana Shatalov, Fall 2010
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FACTS:
– A horizontal line has a slope of
.
– A vertical line has
slope.
– A line that slants upward from left to right (y ↑ as x ↑ ) has
slope.
– A line that slants downward from left to right (y ↓ as x ↑ ) has
slope.
– Two different lines are parallel if and only if their slopes are equal.
– Two lines are perpendicular if the product of the slopes is −1.
• Equations of lines.
Point-slope form: The equation of the line
that passes through the point (x1 , y1 ) and has
slope m is given by: y − y1 = m(x − x1 ).
y
Slope-Intercept Form: The equation of the
line that has slope m and intersects the y-axis
at the point (0,b) is given by: y = mx + b.
y
x
x
0
0
Horizontal line: y = b.
y
Vertical line: x = a.
y
x
x
0
0
General Form of an Equation of a Line: Ax + By + C = 0, where A, B, and C are constants
and A and B are not both zero.
EXAMPLE 3. A line with a slope of 4 passes through point (2,5). What is the equation of
the line?
c Dr Oksana Shatalov, Fall 2010
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EXAMPLE 4. Find the intercepts for the line y = −x + 4 and graph.
y
x
0
EXAMPLE 5. A line has a slope of
y-intercept of the line?
1
2
and passes through the point (−6, 3). What is the
EXAMPLE 6. Find the equation of a line through the point (0, 5) and parallel to the line
y = − 23 x + 2010.
EXAMPLE 7. Find the equation of the line perpendicular to the line 6x + 3y − 29 = 0 and
passing through the point (2, −3).
c Dr Oksana Shatalov, Fall 2010
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EXAMPLE 8. Find the value of a, if the line going through the points (1, a) and (4, −2) is
parallel to the line passing through the points (4, 9) and (2, a + 2).