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1. Parabola A Visual & Mathematical Exploration

2. Introduction to Parabola A parabola is the graph of a quadratic function. It is a U-shaped curve that can open upward, downward, left, or right. Real-life examples: satellite dishes, car headlights, suspension bridges.

3. Parabolas

4. One of nature's best approximations to parabolas is the path of a projectile.

5. This discovery by Galileo in the 17th century made it possible for cannoneers to work out the kind of path a cannonball would travel if it were hurtled through the air at a specific angle.

6. 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.

7. 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 its 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 in front of the focus for the low beam.

8. Key Parts of a Parabola Vertex – the turning point of the parabola. Axis of Symmetry – line dividing the parabola into two equal parts. Focus – fixed point inside parabola. Directrix – fixed line outside parabola. Latus Rectum – line segment through focus, perpendicular to axis.

9. Standard Equations of Parabola

10. Standard Equations of Parabola Opening Up/Down: y² = 4ax Opening Left/Right: x² = 4ay Vertex form (vertical): y = a(x - h)² + k Vertex form (horizontal): x = a(y - k)² + h

11. Graph Features If a > 0 → parabola opens upward (minimum point). If a < 0 → parabola opens downward (maximum point). The axis of symmetry passes through the vertex. The parabola is symmetric about its axis.

12. Parabolas Standard Equations: p>0 Opens UP Opens RIGHT p<0 Opens DOWN Opens LEFT vertex vertex

13. Example Problem Find the equation of the parabola with vertex at (0,0) and focus at (0,3). Solution: Since vertex is at origin and focus is (0,3), axis is vertical. Equation: x² = 4ay → x² = 12y.

14. Example 2 What is the vertex? How does it open? (-2 , 5) opens down Example 3 What is the vertex? How does it open? (0 , 2) opens right