Algebra II End-of-Course Exam Content Standards—Module: Conic Sections
Successful students will be able to represent, analyze, and model using the circle, ellipse, and hyperbola. In addition parabolas with a horizontal axis are included. There are a variety of types of test items including some that cut across the objectives in this standard and require students to make connections and solve rich contextual problems.
C1. Conic Sections
a. Identify a parabola, circle, ellipse, or hyperbola from its equation, description, or key characteristics.
- Key characteristics of the graph include any of the following that apply: vertices, major and minor axes and their lengths, foci, directrix, eccentricity, asymptotes, and center.
- Determine which conic section is represented by a given quadratic equation.
b. Represent conic sections whose axes are parallel to the x- and y-axes using graphs, verbal statements, and equations. Translate among these representations. Represent the equations of conic sections in multiple forms to extract information about the parabola, circle, ellipse, or hyperbola.
- Determine the graph of a conic section whose axes are parallel to the x- and y-axes from its equation, an equation from its graph, and either the equation or graph from verbal information regarding its position and/or characteristics (e.g. vertices, foci, radius, or major and minor axis length.)
- Translate from the general form for conic sections, Ax2 + By2 + Cx + Dy + E = 0, to the standard form for graphic analysis.
- Communicate understanding of the relationship between the standard algebraic forms and graphical characteristics for parabolas, circles, ellipses, and hyperbolas.
- The standard form of an ellipse
identifies the
center as (h, k) and major axis as the x-axis; it provides sufficient information to determine the location of the vertices and foci and the eccentricity.
c. Describe the effect that changes in the parameters of a particular conic section have on its shape and position.
- Parabolas should be limited to those defined by reference to focus, directrix, and/or eccentricity or to parabolas with horizontal axes, i.e. those that can be represented in the form, x = ay2 + by + c.
- Parabolas in the form y = ax2 + by + c are assessed in the core test in Polynomial and Rational Functions.
d. Recognize, express, and solve problems that can be modeled using conic sections. Interpret their solutions in terms of the context of the problem.
- Some real world applications using conic sections may include satellite dish design, planet or satellite orbits, whispering galleries, tracking systems, parabolic mirrors, telescopes, or long range navigational systems.
- This includes using and interpreting appropriate units of measurement and precision for the given application.