Notes parabolas

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  • Notes parabolas

    1. 1. Properties of Parabolas that open Up or Down
    2. 2. Properties of Parabolas that open Up or Down General Form: y = ax2+bx+c
    3. 3. Properties of Parabolas that open Up or Down General Form: y = ax2+bx+c
    4. 4. Properties of Parabolas that open Up or Down General Form: y = ax2+bx+c Standard form (also called vertex form) is y = a(x - h)2 + k, where the vertex is (h,k)  If a is positive it opens up  If a is negative it opens down  Axis of Symmetry (aos) is x = -b/2a
    5. 5. Properties of Parabolas that open Right or Left
    6. 6. Properties of Parabolas that open Right or Left General Form: x = ay2+by+c
    7. 7. Properties of Parabolas that open Right or Left General Form: x = ay2+by+c
    8. 8. Properties of Parabolas that open Right or Left General Form: x = ay2+by+c Standard Form: x = a(y – k)2+ h, where the vertex is (h,k)  If a is positive it opens right  If a is negative it opens left
    9. 9. Properties of Parabolas that open Right or Left General Form: x = ay2+by+c Standard Form: x = a(y – k)2+ h, where the vertex is (h,k)  If a is positive it opens right  If a is negative it opens left Axis of Symmetry (aos) is y = -b/2a
    10. 10. Differences
    11. 11. Differences  Properties of Parabolas that open Left or Right
    12. 12. Differences  Properties of Parabolas that open Left or Right  x=
    13. 13. Differences  Properties of Parabolas that open Left or Right  x=  aos: y =
    14. 14. Differences  Properties of Parabolas that open Left or Right  x=  aos: y =  vertex is (h,k)
    15. 15. Differences Properties of Parabolas  Properties of Parabolas that open Up or Down: that open Left or Right  y=  x=  aos: x =  aos: y =  Vertex is (h,k)  vertex is (h,k)
    16. 16. Examples :Tell the direction that each parabola opens, the vertex and the axisof symmetry (aos). 2 2 y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1
    17. 17. Examples :Tell the direction that each parabola opens, the vertex and the axisof symmetry (aos). 2 2 y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1 Opens up
    18. 18. Examples :Tell the direction that each parabola opens, the vertex and the axisof symmetry (aos). 2 2 y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1 Opens up Vertex is (-2,-6)  make sure to change the sign of the value in the parenthesis
    19. 19. Examples :Tell the direction that each parabola opens, the vertex and the axisof symmetry (aos). 2 2 y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1 Opens up Vertex is (-2,-6)  make sure to change the sign of the value in the parenthesis aos is x = -2  once you have found the vertex you can just take the x coordinate and that is your aos
    20. 20. Examples :Tell the direction that each parabola opens, the vertex and the axisof symmetry (aos). 2 2 y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1 Opens up  Opens left Vertex is (-2,-6)  make sure to change the sign of the value in the parenthesis aos is x = -2  once you have found the vertex you can just take the x coordinate and that is your aos
    21. 21. Examples :Tell the direction that each parabola opens, the vertex and the axisof symmetry (aos). 2 2 y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1 Opens up  Opens left Vertex is (-2,-6)  vertex: (1,-1)  make sure to change the sign of the value in the parenthesis aos is x = -2  once you have found the vertex you can just take the x coordinate and that is your aos
    22. 22. Examples :Tell the direction that each parabola opens, the vertex and the axisof symmetry (aos). 2 2 y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1 Opens up  Opens left Vertex is (-2,-6)  vertex: (1,-1)  make sure to change the sign of the value in the  The thing to remember parenthesis here is that the y- coordinate is now in the aos is x = -2 parenthesis and the x –  once you have found the coordinate is in the back. vertex you can just take the x coordinate and that is your aos
    23. 23. Examples :Tell the direction that each parabola opens, the vertex and the axisof symmetry (aos). 2 2 y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1 Opens up  Opens left Vertex is (-2,-6)  vertex: (1,-1)  make sure to change the sign of the value in the  The thing to remember parenthesis here is that the y- coordinate is now in the aos is x = -2 parenthesis and the x –  once you have found the coordinate is in the back. vertex you can just take  aos: y = -1 the x coordinate and that is your aos
    24. 24. Putting an Equation in Standard Form
    25. 25. Putting an Equation in Standard Form Complete the square.
    26. 26. Putting an Equation in Standard Form Complete the square. Example: y = x2 - 2x + 4  y = (x2 - 2x + ___ ) + 4 – (a) ___ put (-b/2)2 in the blanks and the value for a in the parenthesis before the last blank  y = (x2 - 2x + (2/2)2 ) + 4 – (1)(2/2)2  y = (x2 -2x + 1) + 4 – (1)(1) now factor the first set of ()  y = (x - 1)2 + 4 – 1  y = (x - 1)2 + 3 now it easy to find the vertex and aos.
    27. 27. Determine the direction in which the following open.Solve for either x or y whichever one is only in the problem once or is notsquared. 2 2 6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0
    28. 28. Determine the direction in which the following open.Solve for either x or y whichever one is only in the problem once or is notsquared. 2 2 6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0  Solve for y  2y = -6x2 – 4x +10  y = -3x2 -2x + 5
    29. 29. Determine the direction in which the following open.Solve for either x or y whichever one is only in the problem once or is notsquared. 2 2 6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0  Solve for y 2y = -6x2 – 4x +10   y = -3x2 -2x + 5  Opens down
    30. 30. Determine the direction in which the following open.Solve for either x or y whichever one is only in the problem once or is notsquared. 2 2 6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0  Solve for y  Solve for x 2y = -6x2 – 4x +10   y = -3x2 -2x + 5  Opens down
    31. 31. Determine the direction in which the following open.Solve for either x or y whichever one is only in the problem once or is notsquared. 2 2 6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0  Solve for y  Solve for x 2y = -6x2 – 4x +10   -5x = 5y2 – 10y -10  y = -3x2 -2x + 5  Opens down
    32. 32. Determine the direction in which the following open.Solve for either x or y whichever one is only in the problem once or is notsquared. 2 2 6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0  Solve for y  Solve for x 2y = -6x2 – 4x +10   -5x = 5y2 – 10y -10  y = -3x2 -2x + 5  x = - y2 + 2y + 2  Opens down
    33. 33. Determine the direction in which the following open.Solve for either x or y whichever one is only in the problem once or is notsquared. 2 2 6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0  Solve for y  Solve for x 2y = -6x2 – 4x +10  -5x = 5y2 – 10y -10   y = -3x2 -2x + 5  x = - y2 + 2y + 2  Opens down  Opens left
    34. 34. Write the Standard Form of the equation with a Vertex at (-1,2) and goes through the point (2, 8).
    35. 35. Write the Standard Form of the equation with a Vertex at (-1,2) and goes through the point (2, 8). Identify h, k , x and y  h = -1, k = 2 these are from the vertex  x = 2, y = 8 these are from the other point
    36. 36. Write the Standard Form of the equation with a Vertex at (-1,2) and goes through the point (2, 8). Identify h, k , x and y  h = -1, k = 2 these are from the vertex  x = 2, y = 8 these are from the other point Plug in what you know  y = a(x - h)2 + k  8 = a(2 – (-1))2 + 2
    37. 37. Write the Standard Form of the equation with a Vertex at (-1,2) and goes through the point (2, 8). Identify h, k , x and y  h = -1, k = 2 these are from the vertex  x = 2, y = 8 these are from the other point Plug in what you know  y = a(x - h)2 + k  8 = a(2 – (-1))2 + 2 Now solve for a  8 = 9a + 2  6 = 9a  6/9 = a or a = ⅔
    38. 38. Write the Standard Form of the equation with a Vertex at (-1,2) and goes through the point (2, 8). Identify h, k , x and y  h = -1, k = 2 these are from the vertex  x = 2, y = 8 these are from the other point Plug in what you know  y = a(x - h)2 + k  8 = a(2 – (-1))2 + 2 Now solve for a  8 = 9a + 2  6 = 9a  6/9 = a or a = ⅔ Write the answer in Standard Form, plugging in a h and k  y = ⅔(x + 1)2 + 2

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