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Circles quiz

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  • What is a conic section, and how is a circle obtained from it?
    A conic section is a shape formed by slicing a cone with a plane; a circle is obtained by slicing the cone horizontally.
  • What property makes a circle unique among shapes?
    All points on a circle are equidistant from its center, which is called the radius.
  • How do you find the center of a circle on a graph when the circle is at the origin?
    The center is at the point (0, 0), which is the origin of the graph.
  • How do you determine the radius of a circle from a graph?
    Start at the center and count the number of units to any point on the circle; this distance is the radius.
  • What is the standard equation of a circle centered at the origin?
    The equation is x^2 + y^2 = r^2, where r is the radius.
  • How does the equation of a circle change if the center is not at the origin?
    The equation becomes (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center.
  • How do you find the center and radius from the equation (x-h)^2 + (y-k)^2 = r^2?
    The center is (h, k) and the radius is the square root of r^2.
  • What steps are involved in graphing a circle from its equation?
    Find the center and radius, plot four points at radius distance in all directions from the center, and connect them with a smooth curve.
  • Why is a circle not considered a function?
    A circle fails the vertical line test because a vertical line can intersect the circle at more than one point.
  • What is the general form of a circle's equation?
    The general form is x^2 + y^2 + ax + by + c = 0.
  • What mathematical process is used to convert a circle's general form to standard form?
    Completing the square is used for both x and y terms to convert to standard form.
  • What is the first step in converting a circle's equation from general to standard form?
    Group all x and y terms together and move constants to the other side of the equation.
  • How do you complete the square for the x or y terms in the general form?
    Take the middle term (b), divide by 2, square it, and add it to both sides of the equation.
  • After completing the square, how do you factor the x and y terms?
    Each becomes (x + b/2)^2 or (y + b/2)^2, depending on the term.
  • How do you graph a circle after converting its equation to standard form?
    Identify the center and radius from the standard form, plot four points at radius distance from the center, and connect them to form the circle.