Parabolas as Loci

Explore focus and directrix, vertical standard form, horizontal standard form.

Lesson 2 of 43 parts Practice Flashcards

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Focus and Directrix

Objective: Explain focus and directrix and use it to solve or interpret a problem.

A parabola is the set of points equidistant from a fixed focus and a directrix line. This definition yields standard equations for vertical or horizontal axes. Yes, along the symmetry axis.

Worked examples

Try itParabolas as Loci — practice 1.1: For (x−2)²=4(5)(y−6), find the focus y-coordinate.Show answer

Step 1: Compare with (x−h)²=4p(y−k). Step 2: Here k=6, p=5. Step 3: The focus y-coordinate is k+p=11.

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Focus and Directrix

Objective: Explain focus and directrix and use it to solve or interpret a problem.

A parabola is the set of points equidistant from a fixed focus and a directrix line. This definition yields standard equations for vertical or horizontal axes. Yes, along the symmetry axis.

Worked examples

Try itParabolas as Loci — practice 1.1: For (x−2)²=4(5)(y−6), find the focus y-coordinate.Show answer

Step 1: Compare with (x−h)²=4p(y−k). Step 2: Here k=6, p=5. Step 3: The focus y-coordinate is k+p=11.

Vertical Standard Form

Objective: Explain vertical standard form and use it to solve or interpret a problem.

(x−h)²=4p(y−k) has vertex (h,k), focus (h,k+p), and directrix y=k−p when p≠0. The sign of p controls opening. The parabola opens downward.

Worked examples

Try itParabolas as Loci — practice 2.1: For (x−3)²=4(8)(y−7), find the directrix y-value.Show answer

Step 1: Here k=7 and p=8. Step 2: The directrix is y=k−p. Step 3: Its y-value is -1.

Horizontal Standard Form

Objective: Explain horizontal standard form and use it to solve or interpret a problem.

(y−k)²=4p(x−h) has horizontal axis, focus (h+p,k), and directrix x=h−p. Match the squared coordinate to orientation. Positive p opens right; negative p opens left.

Worked examples

Try itParabolas as Loci — practice 3.1: For (y−3)²=4(11)(x−4), find the focus x-coordinate.Show answer

Step 1: Compare with (y−k)²=4p(x−h). Step 2: Here h=4, p=11. Step 3: The focus x-coordinate is h+p=15.

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