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Geo 3.6&7 slope

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Reveiws slope from algebra I, then shows how to find lines parallel and perpendicular to a given line through a given point.

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Geo 3.6&7 slope

  1. 1. Review of slope from Algebra IGraphing lines of a known slope and intercept
  2. 2. Slope Review from Algebra I
  3. 3. Slope The ratio that describes the tilt of a line is its slope. To calculate slope, you use this ratio. Slope = (Vertical Change) = Rise (Horizontal Change) Run
  4. 4. Slope Equationm = y2 – y 1 x 2 – x1m is the slopepoints (x1 , y1) & (x2 , y2)
  5. 5. Slope A positive slope rises to the right A negative slope falls to the right
  6. 6. Finding Slope on a Graph Remember: Rise over Run. We’re reading from left to right. So start at the left most point and then figure out how to get to the Rise: 2 Rise: -3 next point. Run: 4 Run: 2 Ratio is 2/4 Ratio is -3/2
  7. 7. Finding Slope from 2 Points You can find the slope of the line using the ratio. slope = difference of y – coordinates difference of x – coordinates. The y-coordinate you use first in the numerator must correspond to the x- coordinate you use first in the denominator.
  8. 8. Slope Equationm = y2 – y 1 x 2 – x1m is the slopepoints (x1 , y1) & (x2 , y2)
  9. 9. Find the slope of the linethrough C(-2, 6) and D(4,3) = difference in y-coordinatesSlope difference in x-coordinates = (3 – 6)  y-coordinates (4 – (-2))  x-coordinatesSlope = -3 / 6 = -1/2Down 1, to the Right 2. Cause of Rise (of –1) over Run (+2).
  10. 10. Find the Slope of the Line through each pair of points: V(8, -1) and Q(0, -7) S(-4, 3) and R(-10, 9)
  11. 11. Find the Slope of the Line through each pair of points: V(8, -1) and Q(0, -7) m = 3/4 S(-4, 3) and R(-10, 9) m = -1 = (-1 / 1) if you need a ratio
  12. 12. Special Cases  Horizontal and Vertical lines are special cases This is a horizontal line. The points are (-3, 2) and (1, 2). Therefore, Y = 2. Find the slope. Slope = (2 – 2) / (1 – (-3) = 0 /4 = 0The slope for a horizontal line (or anything Y = ?) is zero.
  13. 13. Special Cases  Horizontal and Vertical lines are special cases This is a vertical line. The points are (-4, 1) and (-4, 3). Therefore, X = -4. Find the slope.Slope = (1 – 3) / (-4 – (-4) = -2 /0 = UndefinedSlope is, therefore, UNDEFINED for vertical lines. lines
  14. 14. Finding the Equation of a Line
  15. 15. Formats for a Linear Equation Standard Form: ax + by = c Slope-Intercept : y = mx + b Use your properties of algebra to convert between the two (Addition Property, Division Property, etc)
  16. 16. Finding the Equation of aLine  Use your slope equation with any point on the line and the point (x, y)  For example the points C(-2, 6) and D(4, 3) earlier had a slope of -1/2  m = y2 – y1 -1 = y – 6 x2 – x 1 2 x – (-2) 2( y-6 ) = -1 ( x – (-2) ) 2y - 12 = -x +2 y = (-1/2) x + 7
  17. 17. Graphing Lines
  18. 18. Graphing Lines This is the graph of y=(-1/2)x + 3. The slope of the line is (-2/4) or (-1/2). The Y-INTERCEPT of • The CONSTANT in the the line is the point equation is the same as where the line crosses the y-intercept. the Y-AXIS.
  19. 19. Graphing Lines This is the graph of y=(-1/2)x + 3. The slope of the line is (-2/4) or (-1/2). y = (-1/2)x + 3 Slope Y-Intercept always a ratio = ConstantFor whole numbers divide by 1
  20. 20. Using Slope-Intercept Form Using the Slope-Intercept Form, you can graph without having to pick points and make a table. y = mx + b Slope-Intercept Form m = Slope of the line. (Ratio) b = Y-Intercept. (Constant) Linear Equations can always be put in this format. It is like solving for y.
  21. 21. To Graph with y = mx + b1) Start with b. Since b is where the line of the equation hits the y-axis, its your first point. Point = (0, b)2) Take the slope, or m. Starting at b, move along the RISE and RUN of the ratio.3) Where you end up is your second point.4) Connect the two dots with a line. (This is the graph of your linear equation).
  22. 22. Lets Graph Together! y = (-1/3)x + 2
  23. 23. Lets Graph Together! y = (-1/3)x + 21) b = 2 so, plot (0, 2) (0, 2)
  24. 24. Lets Graph Together! y = (-1/3)x + 21) b = 2 so, (0, 2)2) Rise: -1, Run: +3 (0, 2)
  25. 25. Lets Graph Together! y = (-1/3)x + 21) b = 2 so, (0, 2)2) Rise: -1, Run: +33) Graph next dot. (0, 2) (2, 1)
  26. 26. Lets Graph Together! y = (-1/3)x + 21) b = 2 so, (0, 2)2) Rise: -1, Run: +33) Graph next dot4) Connect dots with straight line (0, 2) (2, 1)
  27. 27. Finding Parallel andPerpendicular Lines
  28. 28. Parallel Lines Parallel lines have the same slope Find the equation using the same process we used above with the slope and the new point
  29. 29. Example of Parallel Line Find a line parallel to y = (-1/2)x + 7 through point ( 10, 3) -1 = y – 3 2 x – 102(y – 3) = -1 (x – 10) Cross multiplied2y – 6 = -x + 10 Distributive Property2y = -x +16 Added 6 to both sidesy = (-1/2)x + 8 Divided by 2
  30. 30. Perpendicular Line The slope of a perpendicular line is the negative inverse of the original slope For example, if the original slope is -1/2, the perpendicular slope is 2 (Ratio form 2/1) To find a perpendicular line through a given point, use the perpendicular slope and the given point in the slope equation
  31. 31. Example of PerpendicularLine Find a line perpendicular to y = (-1/2)x + 7 through point ( 10, 3) Perpendicular slope = 2 (same as 2 / 1 ) 2 = y–3 Slope equation 1 x – 101 (y – 3) = 2 (x – 10) Cross multipliedy – 3 = 2x - 20 Distributive Propertyy = 2x - 17 Added 3 to both sides
  32. 32. Practice Even problems in the sets below Textbook p168: even of (12-18, 24, 34-42) Textbook p175: even of (8-16, 24-32, 38-44)

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