The Life Starter
Sign up
Grade 11 · lesson 2 of 4

Read a Focus from a Standard Parabola

The vertical parabola x²=4py consists of points equally distant from focus (0,p) and directrix y=−p.

About 25 minutes, plus any extended practice

Download the grade pack

Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: For x²=8y, find the focus. Candidate answers: (A) (0,2) (B) (0,8) (C) (0,−2) (D) (2,0).
  • Independent check inputs: For x²=12y, find the focus. Candidate answers: (A) (0,12) (B) (0,3) (C) (0,−3) (D) (3,0).

Learn the idea

Circle, parabola, ellipse and hyperbola equations encode geometric features through squared quantities and sign patterns. Standard forms locate centers, focal information, axes or vertices, with care to distinguish full lengths from half lengths. The vertical parabola x²=4py consists of points equally distant from focus (0,p) and directrix y=−p. The coefficient four p determines the focal distance and opening direction.

Here is the method on the supplied teaching case. The standard vertical parabola form x²=4py has focus (0,p) and directrix y=−p. Thus p=8/4=2, and the focus is (0,2).

Divide the y coefficient by four to identify p. A focus is not the vertex: the unshifted vertex here is the origin, and positive p places the focus above it.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: For x²=8y, find the focus. Candidate answers: (A) (0,2) (B) (0,8) (C) (0,−2) (D) (2,0). Solution: The standard vertical parabola form x²=4py has focus (0,p) and directrix y=−p. Thus p=8/4=2, and the focus is (0,2). The final answer is (0,2). Divide the y coefficient by four to identify p. A focus is not the vertex: the unshifted vertex here is the origin, and positive p places the focus above it.

Your turn

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Helpful tips
  • The vertical parabola x²=4py consists of points equally distant from focus (0,p) and directrix y=−p.
  • Divide the y coefficient by four to identify p.
For x²=12y, find the focus.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

(0,3)

The standard vertical parabola form x²=4py has focus (0,p) and directrix y=−p. Thus p=12/4=3, and the focus is (0,3).

Take it further

Changed-input task: For x²=36y, find the focus. Candidate answers: (A) (0,36) (B) (0,9) (C) (0,−9) (D) (9,0). Show the mathematical reason, not only the selected answer. Expected result: (0,9). The standard vertical parabola form x²=4py has focus (0,p) and directrix y=−p. Thus p=36/4=9, and the focus is (0,9).

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. For x²=16y, find the focus.
Need a hint?
  • The vertical parabola x²=4py consists of points equally distant from focus (0,p) and directrix y=−p.
  • Divide the y coefficient by four to identify p.
2. For x²=20y, find the focus.
Need a hint?
  • The vertical parabola x²=4py consists of points equally distant from focus (0,p) and directrix y=−p.
  • Divide the y coefficient by four to identify p.
3. For x²=24y, find the focus.
Need a hint?
  • The vertical parabola x²=4py consists of points equally distant from focus (0,p) and directrix y=−p.
  • Divide the y coefficient by four to identify p.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

Check the subject sources

These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.