1. Geometric Properties of Linear Functions
Lesson 1.5

2. 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"?

3. 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

4. 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

5. 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

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

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

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

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

10. 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

11. 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

12. 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 { }

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

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

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