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Circles quiz

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  • What is the definition of a circle in terms of its points and center?
    A circle is the set of all points equidistant from a center point, with the distance being the radius.
  • What are the two key elements needed to graph a circle?
    You need the center (h, k) and the radius (r) to graph a circle.
  • What is the standard equation for a circle centered at the origin?
    The standard equation is x² + y² = r², 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 = r², where (h, k) is the center.
  • How do you find the radius from the standard form equation of a circle?
    Take the square root of the r² term to find the radius.
  • What is the first step in converting a circle's equation from general form to standard form?
    Group the x's and y's together and move all constants to the other side of the equation.
  • What mathematical process is used to convert a circle's equation from general to standard form?
    Completing the square is used for both the x and y terms.
  • How do you complete the square for a term like x² + bx?
    Divide b by 2, square the result, and add it to both sides of the equation.
  • After completing the square, what is the next step in converting to standard form?
    Factor the completed square expressions for both x and y terms.
  • How can you identify the center and radius from the standard form equation?
    The center is (h, k) from (x - h)² + (y - k)², and the radius is the square root of r².
  • Why is a circle not considered a function?
    A circle fails the vertical line test because a vertical line can intersect it at more than one point.
  • What is the general form of a circle's equation?
    The general form is x² + y² + Dx + Ey + F = 0, where D, E, and F are constants.
  • What do you do with the constants when converting from general to standard form?
    Move all constants to the right side of the equation before completing the square.
  • How do you graph a circle once you have its standard form equation?
    Plot the center, use the radius to mark points up, down, left, and right, then connect them with a smooth curve.
  • What is the process for factoring after completing the square in a circle's equation?
    Rewrite the grouped terms as squared binomials, such as (x + b/2)² or (y + b/2)².