Geometric Properties Geometric Properties of Linear Functionsof Linear Functions

Lesson 1.5Lesson 1.5

22

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"?

33

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

44

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

55

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

66

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

77

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

88

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

99

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

1010

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•

1111

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

1212

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 { }

1313

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

1414

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

1515

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