Today in Precalculus Go over homework Notes: Parabolas with vertex other than (0,0) Homework

When a parabola with vertex (0,0) is translated horizontally h units and vertically k units the vertex becomes (h, k) The translation does NOT change the focal length, the focal width, or the direction the parabola opens.

Parabolas with vertex (h,k) Standard Equation(x-h) 2 =4p(y-k)(y-k) 2 = 4p(x-h) OpensUp if p>0 Down if p<0 To right if p>0 To left if p<0 Focus(h, k+p)(h+p, k) Directrixy = k – px = h – p Axisx = hy = k Focal lengthpp Focal width |4p|

Graphing a parabola by hand Let the focus F of a parabola be (2, -3) and its directrix be y = 4. Sketch and label the focus and directrix of the parabola. directrix F

Graphing a parabola by hand Locate, sketch, and label the axis of the parabola What is its equation? x=2 Label and plot the vertex V of the parabola. Label it by name and coordinates. directrix F axis V(2,0.5)

Graphing a parabola by hand What are the focal length and width of the parabola? focal length = p = -3.5 focal width =|4(-3.5)|=14 directrix F axis V(2,0.5)

Use the focal width to locate, plot, and label the endpoints of a chord of the parabola that parallels the directrix. Sketch the parabola. Which direction does it open? downward What is its equation in standard form? (x – 2) 2 = 4(-3.5)(y –.5) (x – 2) 2 = -14(y –.5) V(2,0.5) directrix F axis (-5,-3) (9,-3)

Graphing a parabola with the calculator (x – 2) 2 = -14(y –.5) Solve for y (x – 2) 2 = -14y + 7 (x – 2) 2 – 7 = -14y

Graphing a parabola with the calculator (y – 3) 2 = 6(x – 4) Solve for y

Example Focus (-5,3) and vertex (-5,6) So p=3, opens downward (x + 5) 2 = 4(3)(y – 3) (x + 5) 2 = 12(y – 3)

Example Vertex (3,5) and directrix y = 7 So opens downward, p = –2 (x – 3 ) 2 = 4(-2)(y – 5) (x + 5) 2 = -8(y – 3)

Example Vertex (-3,3), opens right, focal width = 20 (y – 3 ) 2 = 20(x +3)

Homework Page 641: 3,4,22-43 odd