10.2 Parabolas Where is the focus and directrix compared to the vertex? How do you know what direction a parabola opens? How do you write the equation of a parabola given the focus/directrix? What is the general equation for a parabola?

Parabolas A parabola is defined in terms of a fixed point, called the focus, A parabola is the set of all points P(x,y) in the plane whose distance to the focus focus and a fixed line, called the directrix. equals its distance to the directrix. directrix axis of symmetry

Horizontal Directrix Standard Equation of a parabola with its vertex at the origin is x y D(x, –p) P(x, y) F(0, p) y = –p O x2 = 4py p > 0: opens upward p < 0: opens downward focus: (0, p) directrix: y = –p axis of symmetry: y-axis

Vertical Directrix Standard Equation of a parabola with its vertex at the origin is x y D(x, –p) P(x, y) F(p, 0) x = –p O y2= 4px p > 0: opens right p < 0: opens left focus: (p, 0) directrix: x = –p axis of symmetry: x-axis

Example 1 Graph . Label the vertex, focus, and directrix. y2 = 4px Identify p. -4 -2 2 4 y2 = 4(1)x So, p = 1 Since p > 0, the parabola opens to the right. Vertex: (0,0) Focus: (1,0) Directrix: x = -1

Example 1 Graph . Label the vertex, focus, and directrix. Y2 = 4x y x Use a table to sketch a graph -4 -2 2 4 y x 2 4 -2 -4 1 4 1 4

Example 2 Write the standard equation of the parabola with its vertex at the origin and the directrix y = -6. Since the directrix is below the vertex, the parabola opens up Since y = -p and y = -6, p = 6 x2=4(6)y x2 = 24y

Where is the focus and directrix compared to vertex? The focus is a point on the line of symmetry and the directrix is a line below the vertex. The focus and directrix are equidistance from the vertex. How do you know what direction a parabola opens? x2, graph opens up or down, y2, graph opens right or left How do you write the equation of a parabola given the focus/directrix? Find the distance from the focus/directrix to the vertex (p value) and substitute into the equation. What is the general equation for a parabola? x2= 4py (opens up [p>0] or down [p<0]), y2 = 4px (opens right [p>0] or left [p<0])

Assignment p. 598, 16-21, 23-53 odd

10.2 Parabolas, day 2 What does it mean if a parabola has a translated vertex? What general equations can you use for a parabola when the vertex has been translated?

Standard Equation of a Translated Parabola Vertical axis: (x − h)2 = 4p(y − k) vertex: (h, k) focus: (h, k + p) directrix: y = k – p axis of symmetry: x = h

Standard Equation of a Translated Parabola Horizontal axis: (y − k)2 = 4p(x − h) vertex: (h, k) focus: (h + p, k) directrix: x = h - p axis of symmetry: y = k

Example 3 Write the standard equation of the parabola with a focus at F(-3,2) and directrix y = 4. Sketch the info. The parabola opens downward, so the equation is of the form (x − h)2 = 4p(y − k) vertex: (-3,3) h = -3, k = 3 p = -1 (x + 3)2 = 4(−1)(y − 3)

Example 4 Write an equation of a parabola whose vertex is at (−2,1) and whose focus is at (−3, 1). Begin by sketching the parabola. Because the parabola opens to the left, it has the form (y −k)2 = 4p(x − h) Find h and k: The vertex is at (−2,1) so h = −2 and k = 1 Find p: The distance between the vertex (−2,1) and the focus (−3,1) by using the distance formula. p = −1 (y − 1)2 = −4(x + 2)

What does it mean if a parabola has a translated vertex? It means that the vertex of the parabola has been moved from (0,0) to (h,k). What general equations can you use for a parabola when the vertex has been translated? (y-k)2 =4p(x-h) (x-h)2 =4p(y-k)

Assignment p. 598, 38-44 even, 54-68 even p. 628, 15-16, 22, 28