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Understanding Circles: Graphing and Equation Solutions

This lesson focuses on the properties and equations of circles, as well as methods for graphing them. A circle is formed by the intersection of a right cone and a plane perpendicular to its base, and it is not classified as a function. The center of the circle is essential for determining its equation, especially in standard form. You'll learn how to find the radius using the distance formula and how to graph circles using a graphing calculator. Examples include circles centered at the origin and those with different centers.

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Understanding Circles: Graphing and Equation Solutions

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  1. Lesson 91 Making graphs and solving equations of circles

  2. Conic Section

  3. A Circle is formed by the intersection of a right cone and a plane that is perpendicular to the base

  4. It does not pass the vertical line test A circle is NOT a function

  5. If the center is at (0,0), then you can use the distance formula to find the radius Distance formula =r = or+

  6. Equation of a circle with center (0,0) is +=

  7. The equation of a circle must be transformed into 2 functions in order to graph it on a graphing calculator • Isolate y and then enter the positive and negative square roots into the calculator as 2 functions, the graph them together to form a circle Graphing on a graphing calculator

  8. Graph • = so radius is 4 • Plot center at (0,0) • Plot the 4 points that are above, below, left and right of the center • Sketch the circle that passes through the 4 points Graphing circles centered at the origin

  9. 1) = 9 • 2) = 36 Sketch a graph

  10. = 10 • y= • Graph as 2 separate functions • y= and y= Graph- to keep the circle from looking distorted use ZOOM square

  11. Graph on calculator

  12. The equation of a circle with center (h,k) and radius r is • = • In order to graph a circle you must have the center and the radius Standard form of an equation of a circle

  13. Sketch the graph of • radius = 3 center = (-2,1) • Plot the center • plot the points 3 units above, below , left, and right of the center • Sketch the circle that goes through those points Graphing circles not centered at the origin

  14. = 16 • = 11 Graph on calculator

  15. Sometimes the center and radius are not explicitly given, so you might have to use the distance formula and/or the midpoint formula to find them. • M = Distance & midpoint formulas

  16. Write the equation of a circle with center (-3, -1) and radius 7 • h = -3 k = -1 and r = 7 • = • = 49 Writing the equation of a circle

  17. Write the equation of the circle with center (-4,5) and radius 5 Write the equation of circles

  18. Write the equation of the circle with center at (-2,4) that contains the point (5,2) • Find the length of the radius by using the distance formula • r = • r= • = • = 53 Write equation of circle

  19. Write equation of circle with center (3,-2) and that contains the point (-4,2) Write equation of circle

  20. Write the equation of the circle that has a diameter whose endpoints are located at (3,1) and (6,3) • Use the midpoint formula to find the center • M= = = ( 4.5, 2) = center • Find the distance between the center and either of the points on the circle • r= = • So = Write equation of circle

  21. Write the equation of the circle that has a diameter whose endpoints are located at (7,5) and (3,3) Write equation of circle

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