Twelfth Grade · Math · week 34

Conic sections: circles and parabolas

Write the standard-form equation of a circle or parabola, complete the square to identify one from a general equation, and find key features (center, radius, vertex, focus, directrix).

A circle is the set of all points a fixed distance (the radius) from a fixed center: (x-h)^2+(y-k)^2=r^2, with center (h,k) and radius r. A general equation like x^2+y^2+Dx+Ey+F=0 hides this standard form until you complete the square on the x-terms and the y-terms separately, adding whatever balancing numbers are needed to both sides.

A parabola is the set of all points equally distant from a fixed point (the focus) and a fixed line (the directrix). In standard form, (y-k)^2=4p(x-h) opens right (p>0) or left (p<0) with vertex (h,k), focus (h+p,k), and directrix x=h-p; (x-h)^2=4p(y-k) opens up or down instead, with focus (h,k+p) and directrix y=k-p. The distance 4p (not p itself) is what you read directly off the equation's coefficient.

📋 For the teacher
Teach it (10–15 min):
  • Complete the square on the x-terms and y-terms as two clearly separate mini-tasks, writing what is added to EACH side explicitly -- rushing this step is where sign errors creep in.
  • Draw a quick sketch of the focus and directrix for every parabola, and remind them the vertex sits exactly halfway between the two.
  • Have them recover 4p from the equation's coefficient as its own labeled step, before dividing by 4 to isolate p itself.
  • Check one point on a finished circle or parabola (like the example's check) to confirm the center/radius or vertex/focus were found correctly.
Talk about it:
  • Why must you complete the square on the x-terms and y-terms SEPARATELY, rather than all at once?
  • Why is the vertex of a parabola always exactly halfway between its focus and its directrix?
  • How can you tell, just from the sign of p, which direction a parabola opens?

Mastery looks like: They complete the square to find a circle's center and radius from a general equation, and identify a parabola's vertex, focus, and directrix from its standard-form equation.

Common stumbles: Forgetting to add the completed-square balancing numbers to BOTH sides of the equation; mixing up which direction a parabola opens based on the sign of p.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
← Back to todayAll coursesYear planner