1. Warm up Find the coordinates of the center, the foci, vertices and the equation of the asymptotes for

2. Lesson 10-5 Parabolas To use and determine the standard and general forms of the equation of a parabola To graph parabolas

3. A parabola is the collection of all points P in the plane that are the same distance from a fixed point F as they are from a fixed line D. The point F is called the focus of the parabola, and the line D is its directrix. As a result, a parabola is the set of points P for which d(F, P) = d(P, D) Let's sort out this definition by looking at a graph: 2-7-6-5-4-3-21573 0468 focus directrix Take a line segment perpendicular to the directrix and intersect with a line segment from the focus of the same length. This will be a point on the parabola and will be the same distance from each. by symmetry we can get the other half

4. Based on this definition and using the distance formula we can get a formula for the equation of a parabola with a vertex at the origin that opens left or right a is the distance from the vertex to the focus (or opposite way for directrix) If the coefficient on x is positive the parabola opens to the right If the coefficient on x is negative the parabola opens to the left 2 -7-7 -6-6 -5 -4-4 -3-3 -2-2 -1 1573 0468 a a The equation for the parabola shown is: The parabola opens to the right and the vertex is 3 away from the focus.

5. For a parabola with the axis of symmetry parallel to the y-axis and vertex at (h, k), the standard form is … The equation of the axis of symmetry is x = h. The coordinates of the focus are (h, k + p). The equation of the directrix is y = k - p. When p is positive, the parabola opens upward. When p is negative, the parabola opens downward. (x - h) 2 = 4p(y - k) The Standard Form of the Equation with Vertex (h, k)

6. For a parabola with an axis of symmetry parallel to the x-axis and a vertex at (h, k), the standard form is: The equation of the axis of symmetry is y = k. The coordinates of the focus are (h + p, k). The equation of the directrix is x = h - p. (y - k) 2 = 4p(x - h) When p is negative, the parabola opens to the left. When p is positive, the parabola opens to the right. The Standard Form of the Equation with Vertex (h, k)

7. Our parabola may have horizontal and/or vertical transformations. This would translate the vertex from the origin to some other place. The equations for these parabolas are the same but h is the horizontal shift and k the vertical shift: opens up The vertex will be at (h, k) opens down opens right opens left

8. (-1, 2) Let's try one: Opens? y is squared and 8 is positive so right. Vertex? It is shifted to the left one and up 2 (set (x + 1) = 0 and get x = -1 and set (y - 2) = 0 and get y = 2). Vertex is (-1, 2) Focus? 4a = 8 so a = 2. Focus is 2 away from vertex in direction parabola opens. (1, 2) This line segment parallel to the directrix thru the focus is number in front of parenthesis which is 8, so 4 each way from focus. (1, 6) (1, -2) Directrix? "a" away from the vertex so 2 away in opposite direction of focus. x = -3

9. The equation we are given may not be in standard form and we'll have to do some algebraic manipulation to get it that way. Since y is squared, we'll complete the square on the y's and get the x term to other side. middle coefficient divided by 2 and squared 1 1 must add to this side too to keep equation = factor Now we have it in standard form we can find the vertex, focus, directrix and graph.

10. (-3/4, -1) Opens? Right (y squared & no negative) Vertex? opposites of these values (-1, -1) Focus? 4a = 1 so a = 1/4 segment thru the focus 1 1, so 1/2 each way Since the focus was at (-3/4, 1), to get the ends of the segment, we'd need to increase the y value of the focus by 1/2 and then decrease the y value by 1/2. (look at the picture to determine this). (-3/4, -1/2) (-3/4, -3/2) Directrix? 1/4 away from vertex x = -5/4

12. Practice For the equation, find the coordinates of the focus and the vertex and the equations of the directrix and the axis of symmetry.

13. Practice Graph the equation of the parabola

14. Practice Write the equation in standard form.

15. Light or sound waves collected by a parabola will be reflected by the curve through the focus of the parabola, as shown in the figure. Waves emitted from the focus will be reflected out parallel to the axis of symmetry of a parabola. This property is used in communications technology. There are many applications that involve parabolas. One is paraboloids of revolution. This is taking a parabola and revolving it to form "a dish".