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Completing the Square

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Honors Pre-Calculus

Definition

Completing the square is a mathematical technique used to transform a quadratic expression into a perfect square form. This process involves manipulating the coefficients and constants of the quadratic equation to isolate the variable and express the equation in a more simplified, standard form.

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5 Must Know Facts For Your Next Test

  1. Completing the square is a crucial technique for finding the vertex of a quadratic function, which is essential for determining the domain and range.
  2. The process of completing the square involves adding and subtracting a constant to both sides of the equation to transform the quadratic expression into a perfect square form.
  3. Completing the square is often used to convert a quadratic function from standard form to vertex form, which can provide valuable insights into the function's behavior.
  4. In the context of rotating the axes of a quadratic function, completing the square is a fundamental step in determining the new equation and the angle of rotation.
  5. The ability to complete the square is a prerequisite for understanding more advanced topics in mathematics, such as conic sections and complex number systems.

Review Questions

  • Explain how completing the square can be used to determine the domain and range of a quadratic function.
    • Completing the square is an important technique for transforming a quadratic function into vertex form, $f(x) = a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. By identifying the vertex, you can determine the domain and range of the function. The domain is the set of all $x$ values for which the function is defined, often represented as $[h - \\sqrt{-k/a}, h + \\sqrt{-k/a}]$. The range is the set of all possible $y$ values the function can take, which is $[k, \\infty)$ if $a > 0$ or $(- \\infty, k]$ if $a < 0$.
  • Describe the process of completing the square and how it can be used to convert a quadratic function from standard form to vertex form.
    • To complete the square, you start with a quadratic function in standard form, $f(x) = ax^2 + bx + c$. The goal is to manipulate the expression to isolate the variable and express the function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. This is done by adding and subtracting the term $(\\frac{b}{2a})^2$ to both sides of the equation, which effectively creates a perfect square on the left-hand side. The resulting equation can then be simplified to the vertex form, $f(x) = a(x - h)^2 + k$, where $h = -\\frac{b}{2a}$ and $k = c - \\frac{b^2}{4a}$. This transformation provides valuable insights into the function's behavior, such as the location of the vertex, the axis of symmetry, and the function's range.
  • Explain how completing the square is used in the context of rotating the axes of a quadratic function, and discuss the significance of this process.
    • When rotating the axes of a quadratic function, completing the square is a crucial step in determining the new equation and the angle of rotation. By expressing the function in standard form, $f(x) = ax^2 + bx + c$, you can complete the square to transform it into vertex form, $f(x) = a(x - h)^2 + k$. This allows you to identify the coordinates of the vertex $(h, k)$ and the angle of rotation, $\\theta = \\tan^{-1}(-b/a)$. The new equation in the rotated coordinate system can then be written as $f(x, y) = A(x - h\\cos\\theta - y\\sin\\theta)^2 + B(x - h\\cos\\theta + y\\sin\\theta)^2 + k$, where $A$ and $B$ are the coefficients of the new equation. Understanding this process is essential for working with conic sections and other advanced topics in mathematics that involve the transformation of coordinate systems.
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