Copyright © Cengage Learning. All rights reserved. Pre-Calculus Honors 9.1: Circles and Parabolas HW: p.643 (8-24 even, 28, 30, 36, 42)

2 Conics A conic section (or simply conic) is the intersection of a plane and a double-napped cone. Notice in Figure 9.1 that in the formation of the four basic conics, the intersecting plane does not pass through the vertex of the cone. Figure 9.1 Basic Conics

3 Conics The definition of a circle is the collection of all points (x, y) that are equidistant from a fixed point (h, k) leads to the standard equation of a circle (x – h) 2 + (y – k) 2 = r 2. Equation of circle

4 Example 1 – Finding the Standard Equation of a Circle The point (1, 4) is on a circle whose center is at (–2, –3), as shown in Figure 9.4. Write the standard form of the equation of the circle. Figure 9.4

5 Example 2 – Sketching a Circle Sketch the circle given by the equation x 2 – 6x + y 2 – 2y + 6 = 0 and identify its center and radius. Solution: Begin by writing the equation in standard form. x 2 – 6x + y 2 – 2y + 6 = 0 (x 2 – 6x + __) + (y 2 – 2y + __) = –6 + __ + __ Complete the squares In this form, you can identify the center and radius of the circle and then sketch the graph. Write in standard form

6 Example 2 – Solution Now graph the function. cont’d

7 Practice with circles. 1.Write the equation of a circle in standard form with center = (3, 7) and point on the circle = (1, 0). 2.Write the equation of a circle in standard form with center = (-3, -1) and diameter = Identify the center and radius of the circle, then graph. 3.(x – 3) 2 + y 2 = 8 4.x 2 – 14x + y 2 + 8y + 40 = 0 cont’d

8 Do Now Graph 9x 2 + 9y x – 36y + 17 = 0

9 Copyright © Cengage Learning. All rights reserved. Pre-Calculus Honors 9.1: Circles and Parabolas HW: p.644 (70 and 76: write in standard form), (62, 66, 68, 78: graph labeling the vertex and 2 additional points, determine domain and range).

10 Parabolas Vertex: (h, k); there are other characteristics of a parabola we are not going to find and sketch in the graph (directrix and focus). opens up or down b/c x is squared opens right or left b/c y is squared

11 Parabolas Rewrite the parabola in standard form:

12 Parabolas Solution: Convert to standard form by completing the square. –2y = x 2 + 2x – 1 1 – 2y = x 2 + 2x Create a coefficient of 1 for x 2 Isolate the x 2 and bx term __ + 1 – 2y = x 2 + 2x + __ 2 – 2y = x 2 + 2x + 1 –2(y – 1) = (x + 1) 2 Complete the square Combine like terms Factor to write in standard form

13 Graph the parabola. Graph the parabola. Label the vertex and 2 additional points on the parabola. Determine the domain and range. 1. y 2 = 3x

14 Graph the parabola. Graph the parabola. Label the vertex and 2 additional points on the parabola. Determine the domain and range. 2. (x + ½) 2 = 4(y – 1)

15 Graph the parabola. Graph the parabola. Label the vertex and 2 additional points on the parabola. Determine the domain and range. 3. y 2 + x + y = 0

16 Graph the parabola. Graph the parabola. Label the vertex and 2 additional points on the parabola. Determine the domain and range. 4. 3x 2 + 6x + y = 4