SLIDE 1
Section1.4
Equations of Lines and Modeling
Section1.4 Equations of Lines and Modeling FindingtheEquationof - - PowerPoint PPT Presentation
Section1.4 Equations of Lines and Modeling FindingtheEquationof - - PowerPoint PPT Presentation
Section1.4 Equations of Lines and Modeling FindingtheEquationof Line Point-Slope Form If you know the slope and any point on a line, you can write the equation in point-slope form: y y 1 = m ( x x 1 ) Point-Slope Form If you know the
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Point-Slope Form
If you know the slope and any point on a line, you can write the equation in point-slope form: y − y1 = m(x − x1)
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Point-Slope Form
If you know the slope and any point on a line, you can write the equation in point-slope form: y − y1 = m(x − x1) m is the slope of the line rise
run
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Point-Slope Form
If you know the slope and any point on a line, you can write the equation in point-slope form: y − y1 = m(x − x1) m is the slope of the line rise
run
- (x1, y1) is the point
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Examples
- 1. Determine the
slope-intercept form of the equation for the following graph:
−2 −1 1 2 3 4 5 6 7 −4 −3 −2 −1 1
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Examples
- 1. Determine the
slope-intercept form of the equation for the following graph:
−2 −1 1 2 3 4 5 6 7 −4 −3 −2 −1 1
y = 1
2x − 3
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Examples
- 1. Determine the
slope-intercept form of the equation for the following graph:
−2 −1 1 2 3 4 5 6 7 −4 −3 −2 −1 1
y = 1
2x − 3
- 2. Determine the slope-intercept
form of the line which has a slope of −4 and a y-intercept
- f
- 0, 2
3
- .
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Examples
- 1. Determine the
slope-intercept form of the equation for the following graph:
−2 −1 1 2 3 4 5 6 7 −4 −3 −2 −1 1
y = 1
2x − 3
- 2. Determine the slope-intercept
form of the line which has a slope of −4 and a y-intercept
- f
- 0, 2
3
- .
y = −4x + 2
3
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Examples
- 1. Determine the
slope-intercept form of the equation for the following graph:
−2 −1 1 2 3 4 5 6 7 −4 −3 −2 −1 1
y = 1
2x − 3
- 2. Determine the slope-intercept
form of the line which has a slope of −4 and a y-intercept
- f
- 0, 2
3
- .
y = −4x + 2
3
- 3. Find the standard form of a line
with a slope of 3
4 and which
passes through the point (−1, 4).
SLIDE 11
Examples
- 1. Determine the
slope-intercept form of the equation for the following graph:
−2 −1 1 2 3 4 5 6 7 −4 −3 −2 −1 1
y = 1
2x − 3
- 2. Determine the slope-intercept
form of the line which has a slope of −4 and a y-intercept
- f
- 0, 2
3
- .
y = −4x + 2
3
- 3. Find the standard form of a line
with a slope of 3
4 and which
passes through the point (−1, 4). −3x + 4y = 19
- r
3x − 4y = −19
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Examples (continued)
- 4. Find the slope-intercept form of a line which passes through the
points (−3, 1) and (4, 2).
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Examples (continued)
- 4. Find the slope-intercept form of a line which passes through the
points (−3, 1) and (4, 2). y = 1
7x + 10 7
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Examples (continued)
- 4. Find the slope-intercept form of a line which passes through the
points (−3, 1) and (4, 2). y = 1
7x + 10 7
- 5. Find the formula for the linear function g(x), where g(2) = 1 and
g(−1) = 5.
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Examples (continued)
- 4. Find the slope-intercept form of a line which passes through the
points (−3, 1) and (4, 2). y = 1
7x + 10 7
- 5. Find the formula for the linear function g(x), where g(2) = 1 and
g(−1) = 5. g(x) = − 4
3x + 11 3
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ParallelandPerpendicular Lines
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Parallel Lines
Parallel lines are lines which run in the same direction and never cross. Two lines with defined slope are parallel when they have the same slope. m1 = m2
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Parallel Lines
Parallel lines are lines which run in the same direction and never cross. Two lines with defined slope are parallel when they have the same slope. m1 = m2 Two vertical lines (which don’t have a slope) are parallel.
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Perpendicular Lines
Perpendicular lines are lines which cross at exactly a 90◦ angle. Two lines with defined slope are perpendicular when their slopes are negative reciprocals of each other. m1 = − 1
m2
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Perpendicular Lines
Perpendicular lines are lines which cross at exactly a 90◦ angle. Two lines with defined slope are perpendicular when their slopes are negative reciprocals of each other. m1 = − 1
m2
Vertical lines (which don’t have a slope) are perpendicular to horizontal lines.
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Examples
- 1. Are the following pairs of lines parallel, perpendicular, or neither?
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Examples
- 1. Are the following pairs of lines parallel, perpendicular, or neither?
(a) y = 26
3 x − 11;
y = − 3
26x + 15
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Examples
- 1. Are the following pairs of lines parallel, perpendicular, or neither?
(a) y = 26
3 x − 11;
y = − 3
26x + 15
Perpendicular
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Examples
- 1. Are the following pairs of lines parallel, perpendicular, or neither?
(a) y = 26
3 x − 11;
y = − 3
26x + 15
Perpendicular (b) x + 2y = 5; 2x + 4y = 17
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Examples
- 1. Are the following pairs of lines parallel, perpendicular, or neither?
(a) y = 26
3 x − 11;
y = − 3
26x + 15
Perpendicular (b) x + 2y = 5; 2x + 4y = 17 Parallel
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Examples
- 1. Are the following pairs of lines parallel, perpendicular, or neither?
(a) y = 26
3 x − 11;
y = − 3
26x + 15
Perpendicular (b) x + 2y = 5; 2x + 4y = 17 Parallel (c) x = 3; y = 6
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Examples
- 1. Are the following pairs of lines parallel, perpendicular, or neither?
(a) y = 26
3 x − 11;
y = − 3
26x + 15
Perpendicular (b) x + 2y = 5; 2x + 4y = 17 Parallel (c) x = 3; y = 6 Perpendicular
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Examples
- 1. Are the following pairs of lines parallel, perpendicular, or neither?
(a) y = 26
3 x − 11;
y = − 3
26x + 15
Perpendicular (b) x + 2y = 5; 2x + 4y = 17 Parallel (c) x = 3; y = 6 Perpendicular (d) y = −3x + 1; y = − 1
3x + 1
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Examples
- 1. Are the following pairs of lines parallel, perpendicular, or neither?
(a) y = 26
3 x − 11;
y = − 3
26x + 15
Perpendicular (b) x + 2y = 5; 2x + 4y = 17 Parallel (c) x = 3; y = 6 Perpendicular (d) y = −3x + 1; y = − 1
3x + 1
Neither
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Examples (continued)
- 2. Find the equation of the line which is parallel to 2x − y = 6 which
passes through the point (−1, 3).
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Examples (continued)
- 2. Find the equation of the line which is parallel to 2x − y = 6 which
passes through the point (−1, 3). y = 2x + 5
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Examples (continued)
- 2. Find the equation of the line which is parallel to 2x − y = 6 which
passes through the point (−1, 3). y = 2x + 5
- 3. Find the equation of the line which is perpendicular to
−3x + 2y = 1 which passes through the point (3, 5).
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Examples (continued)
- 2. Find the equation of the line which is parallel to 2x − y = 6 which
passes through the point (−1, 3). y = 2x + 5
- 3. Find the equation of the line which is perpendicular to
−3x + 2y = 1 which passes through the point (3, 5). y = − 2
3x + 7
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Examples (continued)
- 2. Find the equation of the line which is parallel to 2x − y = 6 which
passes through the point (−1, 3). y = 2x + 5
- 3. Find the equation of the line which is perpendicular to
−3x + 2y = 1 which passes through the point (3, 5). y = − 2
3x + 7
- 4. Find a so that the line which passes through (a, 4) and (−1, 3) is
parallel to line y = −3x + 6.
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Examples (continued)
- 2. Find the equation of the line which is parallel to 2x − y = 6 which
passes through the point (−1, 3). y = 2x + 5
- 3. Find the equation of the line which is perpendicular to
−3x + 2y = 1 which passes through the point (3, 5). y = − 2
3x + 7
- 4. Find a so that the line which passes through (a, 4) and (−1, 3) is
parallel to line y = −3x + 6. a = − 4
3