1. INTRO TO CONIC SECTIONS

2. IT ALL DEPENDS ON HOW YOU SLICE IT! Start with a cone:

3. IF WE SLICE THE CONE, PARALLEL TO THE BASE, WHAT DO WE GET? A Circle!

4. IF WE SLICE THE CONE AT AN ANGLE, WHAT DO WE GET NOW? An Ellipse!

5. IF WE JUST TAKE A SLICE FROM THE LATERAL FACE OF THE CONE, WHAT DO WE GET? A Parabola!

6. FINALLY, LET’S TAKE A SLICE FROM THE LATERAL FACE, PERPENDICULAR TO THE BASE A Hyperbola!

7. THESE SHAPES ARE CALLED CONIC SECTIONS. WE CAN USE ALGEBRA TO DESCRIBE THE EQUATIONS AND GRAPHS OF THESE SHAPES.

8. 10-2 PARABOLAS

9. Consider a parabola with equation y = ax² 1. If a is positive, then the parabola opens up 2. If a is negative, then the parabola opens down Consider a parabola with equation x = ay ² 3. If a is positive, then the parabola opens right 4. If a is negative, then the parabola opens left

10. LET’S REVIEW PARTS OF A PARABOLA This is the y-intercept, It is where the parabola crosses the y-axis This is the vertex, V (h, k) This is the called the axis of symmetry, a.o.s. Here a.o.s. is the line x = 2 These are the roots Roots are also called: -zeros -solutions - x-intercepts The Vertex Form of a Parabola looks like this: y = a(x-h)² + k ‘a’ describes how wide or narrow the parabola will be.

11. NEW VOCABULARY A parabola is a set of all points that are the same distance from a fixed line and a fixed point not on the line The fixed point (F) is called the focus of the parabola. The fixed line (L) is called the directrix. The focus  A point in the arc of the parabola such that all points on the parabola are equal distance away from the focus and the directrix. The Focus always lies on the axis of symmetry C is the distance between the focus and the vertex c

12. Write an equation for a parabola with a vertex at the origin and a focus at (0, –7). EXAMPLE Step 1:Determine the orientation of the parabola. Does it point up or down? Make a sketch. Since the focus is located below the vertex, the parabola must open downward. Use y = ax 2. where a = negative value. Step 2:Find a.

13. WRITE AN EQUATION FOR A PARABOLA WITH A VERTEX AT THE ORIGIN AND A FOCUS AT (0, –7). An equation for the parabola is y = – x 2. 1 28 Since the parabola opens downward, a is negative. So a = –. 1 28 | a | =Note: c is the distance from the vertex to the focus. =Since the focus is a distance of 7 units from the vertex, c = 7. = 1 4(7) 1 28 1 4c To find a, we use the formula:

14. WRITING EQUATIONS GIVEN … WRITE THE EQ GIVEN VERTEX(0,0) AND FOCUS(0,-2) Draw a graph with given info Use given info to get measurements C = distance from Vertex to Focus, so c = 2 (4-2 of the vertical coordinates) Also, parabola will open down because the focus is below the vertex Use y = -ax² form since opens down To find a, use formula a = 1/4c Therefore a = 1/8 Plug into formula Y = -1/8 x² V(0,0) F(0,-2) C = 2

15. WRITING EQUATIONS GIVEN … WRITE THE EQ GIVEN F(3,0) AND DIRECTRIX IS X = -3 Draw a graph with given info Use given info to get measurements Vertex is in middle of directrix and focus. The distance from the directrix to the focus is 6 units. That means V = (0, 0) C = distance from V to F, so c = 3 Also, parabola will open right Use x = ay² form To find a, use formula a = 1/4c Therefore a = 1/12 Plug into formula x = 1/12 y² F(3,0) C = 3 X = -3 Vertex is in middle of directrix and F So V = (0, 0)

16. Write an equation for a graph that is the set of all points in the plane that are equidistant from point F(0, 1) and the line y = –1. LET’S TRY ONE An equation for a graph that is the set of all points in the plane that are equidistant from the point F (0, 1) and the line y = –1 is y = x 2. 1414

17. Identify the focus and directrix of the graph of the equation x = – y 2. EXAMPLE The parabola is of the form x = ay 2, so the vertex is at the origin and the parabola has a horizontal axis of symmetry. Since a < 0, the parabola opens to the left. 4c = 8 The focus is at (–2, 0). The equation of the directrix is x = 2. c = 2 1818 | a | = 1 4c | – | = 1 4c 1818