Parabolas as Loci

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

Lesson 4 of 63 parts Practice Flashcards

1/3

Focus and Directrix

Objective: Explain focus and directrix, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$(x−h)²=4p(y−k); focus=(h,k+p)$

Worked examples

Try itFor (x−3)²=4(7)(y−5), find the focus y-coordinate.Show answer

Step 1: Compare with (x−h)²=4p(y−k). Step 2: Here k=5, p=7. Step 3: The focus y-coordinate is k+p=12.

Swipe sideways to move between sections

Focus and Directrix

Objective: Explain focus and directrix, apply the rule, and justify relevant restrictions.

Formulas

$(x−h)²=4p(y−k); focus=(h,k+p)$

Worked examples

Try itFor (x−3)²=4(7)(y−5), find the focus y-coordinate.Show answer

Step 1: Compare with (x−h)²=4p(y−k). Step 2: Here k=5, p=7. Step 3: The focus y-coordinate is k+p=12.

Vertical Standard Form

Objective: Explain vertical standard form, apply the rule, and justify relevant restrictions.

(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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$(x−h)²=4p(y−k); directrix y=k−p$

Worked examples

Try itFor (x−3)²=4(7)(y−5), find the directrix y-value.Show answer

Step 1: Here k=5 and p=7. Step 2: The directrix is y=k−p. Step 3: Its y-value is -2.

Horizontal Standard Form

Objective: Explain horizontal standard form, apply the rule, and justify relevant restrictions.

(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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$(y−k)²=4p(x−h); focus=(h+p,k)$

Worked examples

Try itFor (y−5)²=4(7)(x−3), find the focus x-coordinate.Show answer

Step 1: Compare with (y−k)²=4p(x−h). Step 2: Here h=3, p=7. Step 3: The focus x-coordinate is h+p=10.

You finished the reading!

Lock it in with a few practice questions, then move on to the next lesson.

Practice Next lesson

Previous lessonCircles Next lessonEllipses