Acc. Pre Calculus Day 5 - Parabolas Notesheet PARABOLAS

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1 Acc. Pre Calculus Day 5 - Parabolas Notesheet Name Date Block 1) Complete these truths about parabolas: * Parabolas are - shaped. PARABOLAS * Parabolas have a line of. * Parabolas are the graphs of functions. * We know a parabola will be upside down when 2) Write down the actual definition of a parabola. A PARABOLA is 4) There are 5 main features of a parabola. Write them down here: 5) Label these four parts on each of the graphs below. A = F = V = D = p = (Latus Rectum) = General Equation (Vertical) General Equation (Horizontal) Example: x 2 = 12y * The parabola is vertical since there is an x 2 * The vertex is at (0,0) * The p value is 3 since 4p = 12 * The focus is located at (0, 3) * The directrix is the line y = -3 * The Axis of Symmetry is x = 0 (y-axis)

2 Find the same things for these equations: x 2 = 9y x 2 = 1/8 y a) Vertical or Horizontal (circle one) a) Vertical or Horizontal (circle one) b) p value = b) p value = c) Focus: c) Focus: d) Directrix: d) Directrix: e) Axis : e) Axis: f) latus rectum: f) latus rectum: y 2 = 8x y 2 = 1/16 x a) Vertical or Horizontal (circle one) a) Vertical or Horizontal (circle one) b) p value = b) p value = c) Focus: c) Focus: d) Directrix: d) Directrix: e) Axis : e) Axis: f) latus rectum: f) latus rectum: Which direction do each of these parabolas face? (Up, Down, Left, or Right) a) y 2 = - 12x b) x 2 = 1/7 y c) y 2 = 7x d) x 2 = - 24y Example: The vertex of a parabola is at (0, 0) and its focus is at (4,0). * The p value is 4, since there are 4 units from (0,0) to (4,0). * The parabola faces right, since the focus is to the right of the vertex. * The equation of the parabola is y 2 = 4(4)x = 16x. * The directrix is the line x = -4 ( p units to the left of the vertex) * The Latus Rectum is 4 times the length to the focus (4p), so it has a length of 16. Answer the following using the given characteristics: Vertex = (0,0) Focus = (0, 2.5) Vertex = (0,0) Focus = (-2, 0) p value: p value: Directrix: Directrix:

3 Vertex = (0,0) Directrix : x = 3 Vertex = (0,0) Directrix: y = -4 p value: p value: Focus: Focus: Find the following using the given graphs of parabolas: (0,2) (2.5, 0) Focus: Focus: Directrix: Directrix: Equation: Equation: Latus Rectum: Latus Rectum: For Parabolas that do not have a vertex at the origin, the equations are: (y k) 2 = 4p(x h) or (x h) 2 = 4p(y k) (horizontal) (vertical) * The vertex point is located at (h, k) Use the equation to find the following pieces of each parabola. (y 7) 2 = 8(x 2) (x + 3) 2 = 1/12 (y 1) * p value = * p value = * Vertex: * Vertex: * Focus: * Focus: * Directrix: * Directrix: * Latus Rectum: * Latus Rectum:

4 Find the following using the given graphs of parabolas: Vertex: Vertex: Focus: Focus: Directrix: Directrix: Equation: Equation: We can also find the standard forms of parabolas from their general forms. Example: y 2 + 8x 2y 15 = 0 * To put this into standard form, you must get your y terms on one side of your equation and the other terms on the other side. You will then complete the square of the y side to get the standard form. y 2 2y = -8x + 15 (get the y terms on one side) y 2 2y + 1 = -8x ( complete the square) (y 1) 2 = -8x + 16 (simplify) (y 1) 2 = -8(x 2) (simplify to get standard form) Use this process to find the standard form of each parabola. a) x x 6y + 7 = 0 b) y 2 12y + 4x + 4 = 0

5 GRAPH EACH EQUATION: c) x 2 = 12(y 2) d) (y + 2) 2 = 20(x 1) e) (x 4) = 1/8y 2 f) (y + 2) = x 2 + 8x 4