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Warm-up

Warm-up. 1. Find the distance between the points (-1, 4) and (3, 1). 2. Find the midpoint between the points (6, -2) and (2, 9). Graphing Parabolas. Objective: Graph parabolas . Applications.

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Warm-up

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  1. Warm-up 1. Find the distance between the points (-1, 4) and (3, 1). 2. Find the midpointbetween the points (6, -2) and (2, 9).

  2. Graphing Parabolas Objective: Graph parabolas.

  3. Applications • Parabolas exhibit unusual and useful reflective properties. If a light is placed at the focus of a parabolic mirror, the light will be reflected in rays parallel to said axis. In this way a straight beam of light is formed. It is for this reason that parabolic surfaces are used for headlamp reflectors. The bulb is placed at the focus for the high beam and a little above the focus for the low beam.

  4. Applications • The opposite principle is used in the giant mirrors in reflecting telescopes and in antennas used to collect light and radio waves from outer space: the beam comes toward the parabolic surface and is brought into focus at the focal point.

  5. A parabola is the set of all points in a plane that are an equal distance from both a fixed point, the focus, and a fixed line, the directrix.

  6. A parabola is the set of all points P(x, y) in a plane that are an equal distance from both a fixed point, the focus, and a fixed line, the directrix. • A parabola has an axis of symmetry perpendicular to its directrix and that passes through its vertex. • The vertex of a parabola is the midpoint of the perpendicular segment connecting the focus and the directrix.

  7. Previously, you have graphed parabolas with vertical axes of symmetry that open upward or downward. Parabolas may also have horizontal axes of symmetry and may open to the left or right.

  8. Example 1 Sketch the graph of a parabola with focus point (2, 2) and directrix y = -4. Step 1 Plot the focus and draw the directrix. Step 2Find the coordinates of the vertex of the parabola. Vertex at (2, -1) Step 3 Roughly sketch the parabola.

  9. Example 2 Sketch the graph of a parabola with focus point (-6, 5) and directrix x = 4. Step 1 Plot the focus and draw the directrix. Step 2Find the coordinates of the vertex of the parabola. Step 3 Roughly sketch the parabola.

  10. Writing Equations of Parabolas Objective: Write equations of parabolas.

  11. Example 3 Use the Distance Formula to find the equation of a parabola with focus F(2, 4) and directrixy = –4. PF = PD Definition of a parabola. Distance Formula. Substitute (2, 4) for (x1, y1) and (x, –4) for (x2, y2).

  12. Example 3 Continued Simplify. Square both sides. (x – 2)2 + (y – 4)2 = (y + 4)2 Expand. (x – 2)2 + y2 – 8y + 16 = y2 + 8y + 16 Subtract y2 and 16 from both sides. (x – 2)2– 8y = 8y Add 8y to both sides. (x – 2)2 = 16y Solve for y.

  13. Example 4 Use the Distance Formula to find the equation of a parabola with focus F(2, 0) and directrixx = 3. PF = PD Definition of a parabola. Distance Formula. Substitute (2, 0) for (x1, y1) and (3, y) for (x2, y2).

  14. Example 4 Continued Simplify. Square both sides. (x– 2)2 + y2 = (x – 3)2 Expand. x2 – 4x + 4 + y2 = x2 – 6x + 9 Subtract x2 from both sides. –4x + 4 + y2 = –6x + 9 Add 4x and subtract 9 to/from both sides. y2 – 5 = –2x Solve for x.

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