Geometric Properties of Linear Functions Lesson 1.5

Parallel Lines   Parallel lines are infinite lines in the same plane that do not intersect. Note "hyperbolic" lines AB, BC, and DE Which are parallel by the above definition? What about "if two lines are parallel to a third line, then the two lines are parallel to each other"?

Parallel Lines The problem is that this is not what we call a Euclidian system We will be looking at properties of lines in a Euclidian system parallel lines perpendicular lines

Set the style of one of the equations to Thick Parallel Lines Given the two equations y = 2x – 5 y = 2x + 7 Graph both equations How are they the same? How are they different? Set the style of one of the equations to Thick

Parallel Lines Different: where they cross the y-axis Same: The slope Note: they are parallel View Example Parallel lines have the same slope y=2x+7 y=2x-5 Lines with the same slope are parallel

Perpendicular Lines Now consider Graph the lines How are they different How are they the same?

Perpendicular Lines Same: y-intercept is the same Different: slope is different Reset zoom for square Note lines are perpendicular Example

Perpendicular Lines Lines with slopes which are negative reciprocals are perpendicular Perpendicular lines have slopes which are negative reciprocals

Horizontal Lines Try graphing y = 3 What is the slope? How is the line slanted? Horizontal lines have slope of zero y = 0x + 3

Vertical Lines Have the form x = k What happens when we try to graph such a line on the calculator? Think about We say “no slope” or “undefined slope” • k

Intersection of Two Lines Given the two equations We seek an ordered pair (x, y) which satisfies both equations Algebraic solution – set Solve for x Substitute that value back in to one of the equations to solve for y

Intersection of Two Lines Alternative solutions Use the solve() command on calculator solve (y=2x-3.5 and y=-0.5x+4,{x,y}) Graph and ask for intersection Note curly brackets { }

Intersection of Two Lines Alternative solutions Graph and ask for intersection using the spreadsheet Link to IntersectingLines Geogebra file Enter parameters for each line

Intersection of Two Lines Try 3x – y = 17 -2x – 3y = -4 Different rows try different methods Algebraic Solve() command Graph and find intersection

Assignment Lesson 1.5 Page 41 Exercises 1, 3, 5, 6, 9, 11, 15, 17, 25, 29, 31, 33