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Geometric Properties Geometric Properties of Linear Functionsof Linear Functions
Lesson 1.5Lesson 1.5
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Parallel LinesParallel Lines
Parallel lines are Parallel lines are infinite lines in the infinite lines in the same plane that do same plane that do not intersect. not intersect.
Note "hyperbolic" lines AB, BC, and DENote "hyperbolic" lines AB, BC, and DE Which are parallel by the above Which are parallel by the above
definition?definition? What about "if two lines are parallel to What about "if two lines are parallel to
a third line, then the two lines are a third line, then the two lines are parallel to each other"?parallel to each other"?
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Parallel LinesParallel Lines
The problem is thatThe problem is thatthis is not what wethis is not what wecall a Euclidiancall a Euclidiansystemsystem
We will be looking at properties of We will be looking at properties of lines in a Euclidian systemlines in a Euclidian system parallel linesparallel lines perpendicular linesperpendicular lines
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Parallel LinesParallel Lines
Given the two equationsGiven the two equationsy = 2x – 5y = 2x – 5y = 2x + 7y = 2x + 7
Graph both equationsGraph both equations
How are they the same?How are they the same? How are they different?How are they different?
Set the style of one of the
equations to Thick
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Parallel LinesParallel Lines
Different: where they cross the y-Different: where they cross the y-axisaxis
Same: The slopeSame: The slope Note: they are Note: they are parallelparallel
y=2x+
7
y=2x-
5
Parallel lines have the same slope
Parallel lines have the same slope
Lines with the same slope are parallel
Lines with the same slope are parallel
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Perpendicular LinesPerpendicular Lines
Now considerNow consider
Graph the linesGraph the lines How are they differentHow are they different How are they the same?How are they the same?
25
33
52
y x
y x
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Perpendicular LinesPerpendicular Lines
Same:Same: y-intercept is the samey-intercept is the same Different:Different: slope is differentslope is different Reset zoomReset zoom
for for squaresquare
Note lines areNote lines areperpendicularperpendicular
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Perpendicular LinesPerpendicular Lines
Lines with slopes which are Lines with slopes which are negative reciprocalsnegative reciprocals are are perpendicularperpendicular
PerpendicularPerpendicular lines have slopes lines have slopes which are negative reciprocalswhich are negative reciprocals
25
33
52
y x
y x
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Horizontal LinesHorizontal Lines
Try graphing y = 3Try graphing y = 3 What is the slope?What is the slope? How is the line slanted?How is the line slanted?
Horizontal lines have slope of zeroHorizontal lines have slope of zeroy = 0x + 3y = 0x + 3
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Vertical LinesVertical Lines
Have the form Have the form x = x = kk
What happens when we try to What happens when we try to graph such a line on the calculator?graph such a line on the calculator?
Think aboutThink about
We say “no slope” or “undefined We say “no slope” or “undefined slope”slope”
1 2
1 2 0
y y n
x x
k•
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Intersection of Two Intersection of Two LinesLines Given the two equationsGiven the two equations
We seek an ordered pair (x, y) We seek an ordered pair (x, y) which satisfies which satisfies bothboth equations equations
Algebraic solution – setAlgebraic solution – set Solve for xSolve for x Substitute that value back in to one Substitute that value back in to one
of the equations to solve for yof the equations to solve for y
y=2x+3.5
1y=- 4
2x
12x+3.5=- 4
2x
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Intersection of Two Intersection of Two LinesLines Alternative solutionsAlternative solutions
Use the solve() command on Use the solve() command on calculatorcalculatorsolve (y=2x-3.5 and y=-0.5x+4,solve (y=2x-3.5 and y=-0.5x+4,{x,y}){x,y})
Graph and ask for intersectionGraph and ask for intersection
Note curly brackets { }
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Intersection of Two Intersection of Two LinesLines Alternative solutionsAlternative solutions
Graph and ask for intersectionGraph and ask for intersectionusing the spreadsheetusing the spreadsheet
Link to Link to IntersectingLinesIntersectingLines spreadsheetspreadsheet
Enter parameters for each lineEnter parameters for each liney=2x+3.5
1y=- 4
2x
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Intersection of Two Intersection of Two LinesLines TryTry 3x – y = 173x – y = 17
-2x – 3y = -4-2x – 3y = -4 Different rows try different Different rows try different
methodsmethods AlgebraicAlgebraic Solve() commandSolve() command Graph and find intersectionGraph and find intersection
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AssignmentAssignment
Lesson 1.5Lesson 1.5 Page 41Page 41 ExercisesExercises
1, 3, 5, 6, 9, 11, 15, 17, 25, 29, 1, 3, 5, 6, 9, 11, 15, 17, 25, 29, 31, 3331, 33
