1. Geometric Properties of Linear Functions Lesson 1.5

2. 2 Parallel Lines Parallel lines are infinite lines in the same plane that do not intersect. Parallel lines are infinite lines in the same plane that do not intersect. Note "hyperbolic" lines AB, BC, and DE 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"? What about "if two lines are parallel to a third line, then the two lines are parallel to each other"?

3. 3 Parallel Lines The problem is that this is not what we call a Euclidian system 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 We will be looking at properties of lines in a Euclidian system  parallel lines  perpendicular lines

4. 4 Parallel Lines Given the two equations y = 2x – 5 y = 2x + 7 Given the two equations y = 2x – 5 y = 2x + 7 Graph both equations Graph both equations  How are they the same?  How are they different? Set the style of one of the equations to Thick

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

6. 6 Perpendicular Lines Now consider Now consider Graph the lines Graph the lines  How are they different  How are they the same?

7. 7 Perpendicular Lines Same:y-intercept is the same Same:y-intercept is the same Different:slope is different Different:slope is different Reset zoom for square Reset zoom for square Note lines are perpendicular Note lines are perpendicular

8. 8 Perpendicular Lines Lines with slopes which are negative reciprocals are perpendicular Lines with slopes which are negative reciprocals are perpendicular Perpendicular lines have slopes which are negative reciprocals Perpendicular lines have slopes which are negative reciprocals

9. 9 Horizontal Lines Try graphing y = 3 Try 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 zero y = 0x + 3 Horizontal lines have slope of zero y = 0x + 3

10. 10 Vertical Lines Have the form x = k Have the form x = k What happens when we try to graph such a line on the calculator? What happens when we try to graph such a line on the calculator? Think about Think about We say “no slope” or “undefined slope” We say “no slope” or “undefined slope” k

11. 11 Intersection of Two Lines Given the two equations Given the two equations We seek an ordered pair (x, y) which satisfies both equations We seek an ordered pair (x, y) which satisfies both equations Algebraic solution – set Algebraic solution – set  Solve for x  Substitute that value back in to one of the equations to solve for y

12. 12 Intersection of Two Lines Alternative solutions 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 { }

13. 13 Intersection of Two Lines Alternative solutions Alternative solutions  Graph and ask for intersection using the spreadsheet  Link to IntersectingLines spreadsheet IntersectingLines  Enter parameters for each line

14. 14 Intersection of Two Lines Try3x – y = 17 -2x – 3y = -4 Try3x – y = 17 -2x – 3y = -4 Different rows try different methods Different rows try different methods  Algebraic  Solve() command  Graph and find intersection

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