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Use the formula begin mathsize 12px style open parentheses y space minus space k close parentheses squared space equals space 4 a left parenthesis x space minus space h right parenthesis end style or formula begin mathsize 12px style open parentheses x space minus space h close parentheses squared space equals space 4 a left parenthesis y minus k right parenthesis end style for a parabola to write the equation when given the focus and directrix.

 

Clarifications:

Essential Understandings

Concrete:

  • Identify objects that have the shape of a parabola, half an egg, satellite dish and the path the water is making out of the fountain.

    Image of objects that have the shape of a parabola.
  • Throw an object in the air and draw a picture of the path the object takes. Click Here
Representation:
  • Understand that a parabola is set of all points in a plane which are an equal distance away from a given point and given line (illustrated by the blue lines in the diagram below). The point is called the focus of the parabola, and the line is called the directrix.

    Graph illustrating distance from a given point to a given line.
  • Understand that begin mathsize 12px style open parentheses y space minus space k close parentheses squared space equals space 4 a left parenthesis y space minus space k right parenthesis end style represents the vertical parabola and begin mathsize 12px style open parentheses x space minus space h close parentheses squared space equals space 4 a left parenthesis y minus k right parenthesis end style represents the horizontal parabola.
  • Identify what the variables represent in the formula.
  • Understand that the vertex is the midpoint between the focus and the directrix that can be found using the formula y=k-a where y is the directrix, a is half the distance from the focus to the directrix and k is the y value of the vertex. For example: given a focus of (1,0) and a directrix y=-2, the distance between the focus and the directrix is 2. The distance from the focus to the vertex is 1. The directrix y=-2 is added to the distance from the directrix to the vertex, -2+1=-1. The y value of the vertex =-1. The x value of the vertex is the same as the focus x value, 1. The vertex of the parabola is (1,-1).
  • Understand that given a graph a student can count the distance between the focus and the directrix. When this distance is divided in half it gives you the distance to the vertex, (a).
  • Understand that the vertex is a point represented by an ordered pair (h,k). Understand that the x value from the vertex represents h and the y value from the vertex represents k.
  • Understand that parabolas can open vertically and horizontally. Parabolas that open horizontally have a directrix with an x instead of a y. The formula for the directrix of a horizontal parabola is x=h-a. The y value of the vertex is identical to the y value of the focus.
  • Substitute values into the equation for a parabola. For example: Given a parabola with a focus (1,0) and a directrix of y=-2. The distance between the vertex and the directrix = 1.
    h = 1, k=-1 from the vertex (1,-1)

    begin mathsize 12px style open parentheses y space minus space open parentheses negative 1 close parentheses close parentheses squared space equals space 4 left parenthesis 1 right parenthesis left parenthesis x space minus space 1 right parenthesis end style
    begin mathsize 12px style open parentheses y space plus space 1 close parentheses squared space equals space 4 open parentheses x space minus space 1 close parentheses end style

Access Point #: MAFS.912.G-GPE.1.AP.2a (Archived Access Point)
Access Point Standards

Visit the specific benchmark webpage to find related instructional resources.

Access Point Information
Number:
MAFS.912.G-GPE.1.AP.2a
Category:
Access Points
Date Adopted or Revised:
06/14
Cluster:
Translate between the geometric description and the equation for a conic section. (Geometry - Additional Cluster) (Algebra 2 - Additional Cluster) : 

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Access Point Courses
  • Algebra 2 (#1200330): In Algebra 2, instructional time will emphasize five areas: (1) extending arithmetic operations with algebraic expressions to include radical and rational expressions and polynomial division; (2) graphing and analyzing functions including polynomials, absolute value, radical, rational, exponential and logarithmic; (3) building functions using compositions, inverses and transformations; (4) extending systems of equations and inequalities to include non-linear expressions and (5) developing understanding of the complex number system, including complex numbers as roots of polynomial equations.

    All clarifications stated, whether general or specific to Algebra 2, are expectations for instruction of that benchmark

    Curricular content for all subjects must integrate critical-thinking, problem-solving, and workforce-literacy skills; communication, reading, and writing skills; mathematics skills; collaboration skills; contextual and applied-learning skills; technology-literacy skills; information and media-literacy skills; and civic-engagement skills.

  • Algebra 2 Honors (#1200340): In Algebra 2 Honors, instructional time will emphasize six areas: (1) developing understanding of the complex number system, including complex numbers as roots of polynomial equations; (2) extending arithmetic operations with algebraic expressions to include polynomial division, radical and rational expressions; (3) graphing and analyzing functions including polynomials, absolute value, radical, rational, exponential and logarithmic; (4) extending systems of equations and inequalities to include non-linear expressions; (5)building functions using compositions, inverses and transformations and (6) developing understanding of probability concepts.

    All clarifications stated, whether general or specific to Algebra 2 Honors, are expectations for instruction of that benchmark.

    Curricular content for all subjects must integrate critical-thinking, problem-solving, and workforce-literacy skills; communication, reading, and writing skills; mathematics skills; collaboration skills; contextual and applied-learning skills; technology-literacy skills; information and media-literacy skills; and civic-engagement skills.

  • Precalculus Honors (#1202340): In Precalculus Honors, instructional time will emphasize six areas: (1) extending right triangle trigonometry to unit circle trigonometry and trigonometric functions; (2) extending understanding of functions to trigonometric; (3) developing understanding of conic sections; (4) representing and performing operations with complex numbers and vectors in the coordinate plane; (5) extending understanding of relations in the plane using parametric representations, including polar coordinates and (6) analyzing arithmetic and geometric sequences and series.

    All clarifications stated, whether general or specific to Precalculus Honors, are expectations for instruction of that benchmark.

    Curricular content for all subjects must integrate critical-thinking, problem-solving, and workforce-literacy skills; communication, reading, and writing skills; mathematics skills; collaboration skills; contextual and applied-learning skills; technology-literacy skills; information and media-literacy skills; and civic-engagement skills.

  • Liberal Arts Mathematics (#1207310):
  • Geometry Honors (#1206320): In Geometry Honors, instructional time will emphasize five areas: (1) proving and applying relationships and theorems involving two-dimensional figures using Euclidean geometry and coordinate geometry; (2) establishing congruence and similarity using criteria from Euclidean geometry and using rigid transformations; (3) extending knowledge of geometric measurement to two-dimensional figures and three-dimensional figures; (4) creating and applying equations of circles in the coordinate plane and (5) developing an understanding of right triangle trigonometry.

    All clarifications stated, whether general or specific to Geometry Honors, are expectations for instruction of that benchmark.

    Curricular content for all subjects must integrate critical-thinking, problem-solving, and workforce-literacy skills; communication, reading, and writing skills; mathematics skills; collaboration skills; contextual and applied-learning skills; technology-literacy skills; information and media-literacy skills; and civic-engagement skills.

  • Advanced Topics in Mathematics (formerly 129830A) (#1298310):
  • Algebra 2 for Credit Recovery (#1200335): Special notes: Credit Recovery courses are credit bearing courses with specific content requirements defined by Next Generation Sunshine State Standards and/or Florida Standards. Students enrolled in a Credit Recovery course must have previously attempted the corresponding course (and/or End-of-Course assessment) since the course requirements for the Credit Recovery course are exactly the same as the previously attempted corresponding course. For example, Geometry (1206310) and Geometry for Credit Recovery (1206315) have identical content requirements. It is important to note that Credit Recovery courses are not bound by Section 1003.436(1)(a), Florida Statutes, requiring a minimum of 135 hours of bona fide instruction (120 hours in a school/district implementing block scheduling) in a designed course of study that contains student performance standards, since the students have previously attempted successful completion of the corresponding course. Additionally, Credit Recovery courses should ONLY be used for credit recovery, grade forgiveness, or remediation for students needing to prepare for an End-of-Course assessment retake.
  • Access Algebra 2 (#7912095):
    Access Courses:

    Access courses are for students with the most significant cognitive disabilities. Access courses are designed to provide students access to grade-level general curriculum. Access points are alternate academic achievement standards included in access courses that target the salient content of Florida’s standards. Access points are intentionally designed to academically challenge students with the most significant cognitive disabilities.