Graph circles (Tutorial Only)

Learning Objectives Identify the center and radius of a circle from its standard equation. Write the standard equation of a circle given its center and radius. Graph a circle on the Cartesian plane from its standard equation. Convert the general form of a circle's equation to standard form by completing the square. Graph a circle from its general equation. Determine the equation of a circle given its center and a point on the circle, or the endpoints of a diameter. Ever wonder how your phone's GPS pinpoints your location with such accuracy? It uses the intersection of circles! 🛰️ This tutorial explores the circle, a fundamental conic section. You will learn to work with the standard and general equations of a circle, enabling you to derive key properties like the...

TermDefinitionExample CircleThe set of all points in a plane that are equidistant from a fixed point, called the center.All points that are exactly 5 units away from the point (2, 3) form a circle. Center (h, k)The fixed point from which all points on the circle are equidistant. It is the geometric middle of the circle.In the equation (x - 4)^2 + (y + 1)^2 = 25, the center is at the point (4, -1). Radius (r)The fixed distance from the center to any point on the circle. It must be a positive value.For a circle with the equation x^2 + y^2 = 9, the radius is r = √9 = 3. Standard Form of a Circle's EquationThe equation of a circle written as (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.The equation (x - 1)^2 + (y - 2)^2 = 16 is in standard form. General For...

Standard Form Equation of a Circle (x - h)^2 + (y - k)^2 = r^2 Use this form to directly identify the circle's center (h, k) and its radius, r. This is the most useful form for graphing. Remember to take the opposite sign for h and k from the equation and the square root of the constant term for the radius. Midpoint Formula (for finding the center) (h, k) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) Use this formula to find the center of a circle when you are given the coordinates of the endpoints of a diameter, (x_1, y_1) and (x_2, y_2). Distance Formula (for finding the radius) r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Use this formula to find the radius when you know the center (x_1, y_1) and any point on the circle (x_2, y_2). It is a direct applic...