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Comprehensive Review of Circle and Parabola Equations Centered at the Origin and Beyond

This review covers essential concepts of circle and parabola equations for curves centered at the origin (0,0) and those centered at other points (h, k). Learn about the unique properties of circles, including the center and radius derived from the standard equation. Explore parabolas, their orientations, vertex, focus, and directrix through guided problems and solutions. Engage with sample equations to solidify your understanding of these fundamental geometric shapes as they appear in various forms and contexts.

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Comprehensive Review of Circle and Parabola Equations Centered at the Origin and Beyond

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  1. Circles and ParabolasReview

  2. Equations of the Curves centered at the origin (0,0) • Circle • Parabola or

  3. Equations for Curves not centered at the origin • Circle Centered at (h, k) • Parabola or

  4. Circles Circles are a special type of ellipses. There is a center that is the same distance from every point on the diameter. In the equation the center is at (h, k). The distance from the center to any point on the line is called the radius of the circle. From the equation to find the radius you take the square root of r2.

  5. Parabolas A parabola is a curve that is oriented either up, down, left, or right. The vertex of the parabola is at (h, k). In the equation the h value added or subtracted to x moves the parabola left and right. If you subtract the value of h the parabola moves to the right. If you add the value of h the parabola moves to the left. Parabolas are symmetrical across the line through the vertex of the parabola.

  6. Problem 1Circle • Graph the following equation of a circle • (x- 3)2 + (y- 3)2 = 16 • *Find first before graphing • the center for the circle • the radius for the circle.

  7. Solution to Problem 1 Center: (3,3) Radius: 4

  8. Problem 2Parabola • Graph the following equation of the parabola. • (x + 2)2 = ½ (y – 1) • Determine the following before graphing the equation: • Which way does the parabola open? • The vertex of the parabola. • The focus and the directrix

  9. Solution to Problem 2 Opens up Vertex: (-2, 1) To get the Focus: ½ ÷ 4 = ½ ∙ ¼ = 1/8, so from the vertex (-2, 1) we stay at -2 and add 1/8 to the y coordinate (1). Focus: (-2, 9/8) To get the Directrix: From the vertex we subtract 1/8 from the y coordinate (1). Directrix: y = 7/8 (-2, 9/8) y = 7/8

  10. Problem 3 Click on the correct answer to move to the next problem. What type of object/ curve is given by the equation below? What is the center of the equation? A. Circle Center (-2,-10) B. Circle Center (2,10) C. Parabola Center (2,10) D. Parabola Center (-2,-10)

  11. Problem 4 Click the correct answer to continue. What is the center and the radius of the following circle equation? A. Center (-1,0) Radius = 10 B. Center (0,1) Radius = 100 C. Center (0,1) Radius = 10 D. Center (-1,0) Radius = 100

  12. Begin Homework Pages 89-90 (Math Mate 7 is due next class)

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