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Algebra II

This cheat sheet covers topics such as midpoint and distance formulas, equations of parabolas, and equations of circles. It includes examples and instructions on how to graph various conic sections.

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Algebra II

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  1. Algebra II Chapter 8: Conic Sections

  2. Cheat Sheet • In chapter 8 you are allowed a “cheat sheet” • You are to bring in a tissue box, that has not been opened, and cover it with paper. • You are allowed to write anything on this box that you so choose. • You may use it on your quiz and chapter test. • When we are done with Chapter 8, you must give the tissue box to me. • It is your decision to do this, you may not have any other form a “cheat sheet”

  3. 8.1: Midpoint and Distance Formulas • Find the midpoint of a segment on the coordinate plane • Find the distance between two points on the coordinate plane

  4. The Midpoint Formula • The midpoint is the point in the middle of a segment • Definition: M is the midpoint of PQ if M is between P and Q and PM = MQ. • Formula:

  5. Example 1: • Find the midpoint of each line segment with endpoints at the given coordinates: • (12, 7) and (-2, 11) • (-8, -3) and (10, 9) • (4, 15) and (10, 1) • (-3, -3) and (3, 3)

  6. “Curveball Problem” Example 2: • Segment MN has a midpoint P. If M has coordinates (14, -3) and P has coordinates (-8, 6), what are the coordinates of N? • Circle R has a diameter ST. If R has coordinates (-4, -8) and S has coordinates (1, 4), what are the coordinates of T?

  7. “Curveball Problem” Example 2: • Circle Q has a diameter AB. If A is at (-3,-5) and the center is at (2, 3), find the coordinates of the B.

  8. The Distance Formula • Distance is always a positive number • You can find distance using the Pythagorean Theorem or using a formula derived from it • Formula:

  9. Example 3: • Find the distance between each pair of points with the given coordinates • (3, 7) and (-1, 4) • (-2, -10) and (10, -5) • (6, -6) and (-2, 0)

  10. Example 4: • Rectangle ABCD has vertices A(1, 4), B(3, 1), C(-3, -2), and D(-5, 1). Find the perimeter and area of ABCD • Circle R has diameter ST with endpoints S(4, 5) and T(-2, -3). What are the circumference and are of the circle? (Round to two decimal places)

  11. Summary: • Learn the midpoint and distance formulas • Be able to answer any question that may involve them. • Questions?

  12. 8.2: Parabolas • Write equations of parabola in standard form and vertex form • Graph parabolas

  13. Equations of Parabolas • Standard Form • y = ax2 + bx + c • Vertex Form • y = a(x – h)2 + k

  14. Example 1: • Write y = 3x2 + 24x + 50 in vertex form. Identify the vertex, axis of symmetry, and direction of opening of the parabola.

  15. Example 1: • Write y = -x2 – 2x + 3 in vertex form. Identify the vertex, axis of symmetry, and direction of opening of the parabola.

  16. Graph Parabola • You must always graph: • Vertex • Axis of Symmetry • Five points on the graph (this is to get the shape) • Focus: point in which all points in a parabola are equidistant • Directrix: line that the parabola will never cross

  17. Concept Summary (pg 422)

  18. Example 2: • Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola • y = x2 + 6x – 4 • x = y2 – 8y + 6

  19. Example 2: • Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola • y = 8x – 2x2 + 10 • x = -y2 – 4y – 1

  20. Graph: y = ½(x – 1)2 + 2 Graph: x = -2(y + 1)2 - 3 Example 3:

  21. Classwork/Homework • Workbook • Section 8.1 • 1, 3, 5, 11, 17, 19, 21, 31, 32 • Section 8.2 • 1 – 6 (all)

  22. 8.3: Circles • Write equations of circles • Graph circles

  23. Circle • A circle is the set of all point in a plane that are equidistant from a given point in the plane, called the center. • Equation of a circle: • (x – h)2 + (y – k)2 = r2

  24. Example One: • Write an equation for the circle that satisfies each set of conditions: • Center (8, -3), Radius 6 • Center (5, -6), Radius 4

  25. Example One: • Write an equation for the circle that satisfies each set of conditions: • Center (-5, 2) passes through (-9, 6) • Center (7, 7) passes through (12, 9)

  26. Example One: • Write an equation for the circle that satisfies each set of conditions: • Endpoints of a diameter are (-4, -2) and (8, 4) • Endpoints of a diameter are (-4, 3) and (6, -8)

  27. Graph circles • Make sure the equation is in standard form • Graph the center • Use the length of the radius to graph four points on the circle (up, down, left, right) • Connect the dots to create the circle

  28. Example Two: • Find the center and radius of the circle given the equation. Then graph the circle • (x – 3)2 + y2 = 9

  29. Example Two: • Find the center and radius of the circle given the equation. Then graph the circle • (x – 1)2 + (y + 3)2 = 25

  30. Example Two: • Find the center and radius of the circle given the equation. Then graph the circle • x2 + y2 – 10x + 8y + 16 = 0

  31. Example Two: • Find the center and radius of the circle given the equation. Then graph the circle • x2 + y2 – 4x + 6y = 12

  32. Classwork/Homework • Workbook • Lesson 8.3 • 1 – 13 (all)

  33. Homework Answers: Workbook 8.3 • (x + 4)2 + (y – 2)2 = 64 • x2 + y2 = 16 • (x + ¼)2 + (y + )2 = 50 • (x – 2.5)2 + (y – 4.2)2 = 0.81 • (x + 1)2 + (y + 7)2 = 5 • (x + 9)2 + (y + 12)2 = 74 • (x + 6)2 + (y – 5)2 = 25 • (-3, 0); r = 4 • (0, 0); r = 2 • (-1, -3); r = 6 • (1, -2); r = 4 • (3, 0); r = 3 • (-1, -3); r = 3

  34. Homework Review

  35. 8.4: Ellipses • Write equations of ellipses • Graph ellipses

  36. Ellipse • An ellipse is like an oval. • Every ellipse has two axes of symmetry • Called the major axis and the minor axis • The axes intersect at the center of the ellipse • The major axis is bigger than the minor axis • We use c2 = a2 – b2 to find c • a is always greater b • The equation is always equal to 1

  37. Ellipses Chart (pg 434)

  38. Example One: Graph the ellipse

  39. Your Turn: Graph the ellipse

  40. Example Two: Graph the ellipse

  41. Your Turn: Graph the ellipse

  42. Example Three: Write the equation of the ellipse in the graph:

  43. Your Turn: Write the equation of the ellipse in the graph:

  44. Example Four: Write the equation of the ellipse in the graph:

  45. Your Turn: Write the equation of the ellipse in the graph:

  46. Standard Form Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation: x2 + 4y2 + 24y = -32

  47. Standard Form Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation: 9x2 + 6y2 – 36x + 12y = 12

  48. Classwork

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