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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 of a circle, what do h and k represent?
    h and k represent the x and y coordinates of the center of the circle, respectively.
  • 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, where the center is at (0, 0).
  • What is the first step in converting a circle's general form to standard form?
    Group the x terms together and the y terms together, and move all constants to the other side of the equation.
  • What algebraic process is used to convert the general form of a circle to standard form?
    You use the process of completing the square for both the x and y terms.
  • How do you complete the square for a term like x^2 + bx?
    Take b, divide by 2, square the result, and add it to both sides of the equation.
  • After completing the square, what should you do next to write the equation in standard form?
    Factor the perfect square trinomials for both x and y terms.
  • If the equation is (x - 2)^2 + (y + 1)^2 = 16, what is the center and radius of the circle?
    The center is (2, -1) and the radius is 4.
  • Why is y + 1 written as y - (-1) in standard form?
    Because the standard form uses y - k, so k is -1 when the equation has y + 1.
  • What does the set of all points the same distance from a center describe?
    It describes a circle, where the distance is called the radius.
  • When graphing a circle, how do you use the radius?
    Plot points above, below, left, and right of the center at a distance equal to the radius, then connect them smoothly.
  • If you have x^2 + 2x + y^2 - 4y + 1 = 0, what is the first step to convert to standard form?
    Group x and y terms and move the constant to the other side: x^2 + 2x + y^2 - 4y = -1.
  • How do you find the value to add when completing the square for y^2 - 4y?
    Take -4, divide by 2 to get -2, then square it to get 4, and add 4 to both sides.
  • Once you have (x + 1)^2 + (y - 2)^2 = 4, what are the center and radius?
    The center is (-1, 2) and the radius is 2.