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Graphing Circles quiz

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  • What is the standard form equation of a circle?
    The standard form is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
  • In the standard form equation of a circle, what do h and k represent?
    h and k represent the x and y coordinates of the center of the circle.
  • How do you determine the radius from the standard form equation of a circle?
    The radius is the square root of the number on the right side of the equation, r = √(r^2).
  • What is the equation of a circle centered at the origin?
    The equation is x^2 + y^2 = r^2.
  • How do you graph a circle given its center and radius?
    Plot the center, then mark points r units above, below, left, and right of the center, and connect them with a smooth curve.
  • What is the general form of a circle's equation?
    The general form is x^2 + y^2 + Dx + Ey + F = 0.
  • What algebraic process is used to convert a circle's equation from general to standard form?
    You use completing the square for both x and y terms.
  • What is the first step in converting a general form equation to standard form?
    Group the x terms together, the y terms together, and move the constant to the other side of the equation.
  • How do you complete the square for a term like x^2 + bx?
    Take b, divide by 2, square it, and add it to both sides of the equation.
  • After completing the square, what form do the x and y terms take?
    They become perfect square trinomials, which can be factored as (x + d)^2 or (y + e)^2.
  • If the equation is (x + 1)^2 + (y - 2)^2 = 4, what is the center and radius of the circle?
    The center is (-1, 2) and the radius is 2.
  • Why must you add the same value to both sides when completing the square?
    To maintain the equality of the equation after modifying one side.
  • How do you identify h and k if the equation has (x + a) or (y + b)?
    h is -a and k is -b, since the standard form is (x - h) and (y - k).
  • What does the graph of x^2 + y^2 = 9 look like?
    It is a circle centered at the origin with a radius of 3.
  • What is the purpose of converting a circle's equation to standard form before graphing?
    It makes it easy to identify the center and radius, which are needed to accurately graph the circle.