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Geometric Properties of Linear Functions

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?

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Geometric Properties of Linear Functions

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  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 thatthis is not what wecall a Euclidiansystem • 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 equationsy = 2x – 5y = 2x + 7 • Graph both equations • How are they the same? • How are they different?

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

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

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

  15. Assignment • Lesson 1.5 • Page 41 • Exercises1, 3, 5, 6, 9, 11, 15, 17, 25, 29, 31, 33

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