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Parabolas (part 2)

Learn how to transform parabolas into vertex form, complete the square, factor non-squared terms, and graph accurately. Understand focus points and vertex coordinates.

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Parabolas (part 2)

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  1. Parabolas (part 2) I. Vertex form of a Parabola. A) (x – h)2 = a(y – k) or (y – k)2 = a(x – h) 1) Vertex is at (h , k) [Change the signs] B) You will need to be able to complete the square and factor to get an equation into Vertex form. C) You can find the focus easily from the Vertex form. 1) Divide the “a” by 4. The squared term gets the 0. 2) Add this value to the Vertex point (h , k) to get the focus point. Vertex (x , y) + Focus (0 , #) or (# , 0).

  2. Parabolas (part 2) II. Completing the Square [ ½ the #, square the # ]. A) Complete the square on the squared term only. 1) Make the equation say 1x2 or 1y2. a) Divide/Multiply everything to make coefficient = 1. 2) Move everything except the x2 + bx or the y2 + by terms to the other side of the = sign. 3) Complete the square: ½ the number, square that #. a) x2 + bx  (x + b/2)2 = … + (b/2)2 -- or -- y2 + by  (y + b/2)2 = … + (b/2)2 4) Collect like terms on the non-squared side.

  3. Parabolas (part 2) III. Factoring the Non-Squared Side. A) After you have collected like terms on the non-squared side, factor out the number in front of the non-squared term from both numbers on that side of the = sign. 1) (x – h)2 = ay + b  (x – h)2 = a(y + b/a) B) The equation is now in Vertex form. 1) (x – h)2 = a(y + b/a) or (y – k)2 = a(x + b/a) C) Finding the focus point. 1) Divide the “a” term by 4 ( or multiply by ¼ ). 2) The squared term gets the 0, the non-squared term gets the focus value. 3) Add the focus (0 , #) or (# , 0) to the vertex’s (x , y) to get the coordinates of the focus.

  4. Parabolas (part 2) IV. Graphing the Parabola in Vertex Form. A) If the equation is in Vertex form [ vertex is (h , k) ]. 1) (x – h)2 = a(y – k) or (y – k)2 = a(x – h) B) Ignore (erase) the sideways & up/down shift values. Only keep the “a” term. C) Get the equation into y = ax2 or x = ay2 form. 1) You will have to divide by the “a” term to move it. D) Determine the direction the parabola will open and if it should be taller (a>1 = narrow) or shorter (0>a>1 = wider). Examples: (x – 3)2 = 5(y + 1) (y + 6)2 = ½(x – 4) Ignore the shifts x2 = 5y y2 = ½x Move the “a” term 1/5x2 = y 2y2 = x Determine the shape wide-opens upward narrow-opens right

  5. Parabolas (part 2) Examples: x2 + 4x – 12y + 40 = 0 (x2 + 4x) = 12y – 40 (x + 2)2 = 12y – 40 + (2)2 [ ½ the # , #2 ] (x + 2)2 = 12y – 36 (x + 2)2 = 12(y – 3) Vertex = (–2 , 3) Focus is 12/4 = 3  so y = +3. so Focus = (–2 , 3+3) = (–2 , 6) Eq: 12y = x2 y = 1/12x2

  6. Parabolas (part 2) Examples: 3y2 – 18y + 6x – 12 = 0 [divide everything by 3] y2 – 6y + 2x – 4 = 0 [move non-y terms over] (y2 – 6y) = –2x + 4 (y – 3)2 = –2x + 4 + (3)2 [ ½ the # , #2 ] (y – 3)2 = –2x + 13 (y – 3)2 = –2(x – 13/2) Vertex = (13/2 , 3) or (6.5 , 3) Focus is –2/4 = –½  so x = –.5 so Focus = (6.5 – .5 , 3) = (6 , 3) Eq: –2x = y2 x = –½ y2

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