Chapter 6 – Conic Sections

Activities – Chapter 6

Activity 6a – Completing the Square

This activity is designed to develop the ‘Completing the Square’ technique. You should focus on the technique itself, and not the results. The idea is that many graphs rely on a standard form which give a lot of information about the graph.

Examples include:

  • Equation of a Circle: [latex](x-h)^2 + (y-k)^2 = r^2[/latex], where [latex](h,k)[/latex] is the center of the circle and [latex]r[/latex] is the radius.
  • Equation of a Parabola: [latex]y = a(x-h)^2 + k[/latex], where [latex](h,k)[/latex] is the vertex, and [latex]a[/latex] determines a vertical stretch factor.
  • Equation of an Ellipse: [latex]\displaystyle{\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1}[/latex], where [latex](h,k)[/latex] is the center, and [latex]a[/latex] and [latex]b[/latex] determine the lengths of the major and minor axes.
  • Equation of a Hyperbola: [latex]\displaystyle{\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1}[/latex], where [latex](h,k)[/latex] is the center, and [latex]a[/latex] and [latex]b[/latex] determine the slopes of the asymptotes.

The common theme is that the variables [latex]x[/latex] and [latex]y[/latex] only appear in terms that are squared. There are no linear terms like [latex]3x[/latex] or [latex]-8y[/latex]. The key, therefore, is to rewrite equations so that terms involving [latex]x[/latex] and [latex]y[/latex] can be factored as perfect squares. This is where the name of the technique comes from, rearrange terms to find constants to be added that form a pattern that can be factored; these constants ‘complete’ the square.

Students should focus on the goal equation, which factors will occur, and which constants will produce these factors. The process often goes in reverse, starting with the proposed factors, and working backward to find the constants. Students are best served if they understand the factors, and think about factoring issues, rather than trying to figure out the constants initially.

This activity will enable students to convert the general form of conic sections (circle, ellipse, parabola, hyperbola) to the standard form listed above.

Completing the Square

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Trigonometry by Mike Weimerskirch and the University of Minnesota Board of Regents is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.