Completing the Square to Graph Parabolas

Completing the Square and Graphing Parabolas

Review of Completing the Square

• Completing the square involves creating a perfect square trinomial that can be factored into a binomial squared.
• Example:
  • Initial expression: $x^2 - 10x - 4$
  • Goal: Transform it into $(x - 5)^2$ form.
  • The perfect square requires adding 25 (since $-5 \times -5 = 25$).
  • Steps:
    1. Move the constant term to the right side: $x^2 - 10x = 4$
    2. Calculate the value needed to complete the square: $(\frac{-10}{2})^2 = 25$
    3. Add this value to both sides: $x^2 - 10x + 25 = 4 + 25$
    4. Factor the left side and simplify the right side: $(x - 5)^2 = 29$

Parabolas and Completing the Square

• $x^2$ represents a vertically opening parabola.
• $y^2$ represents a horizontally opening parabola.

Finding Focus, Directrix, and Vertex

• Problem: Find the focus, directrix, and vertex for the parabola $x^2 - 6x + y - 6 = 0$
• Isolate x terms and move y terms to the other side:
  $x^2 - 6x = -y + 6$
• Complete the square for the x terms:
  • Take half of the coefficient of x: $\frac{-6}{2} = -3$
  • Square it: $(-3)^2 = 9$
  • Add 9 to both sides: $x^2 - 6x + 9 = -y + 6 + 9$
  • Rewrite: $(x - 3)^2 = -y + 15$
• Factor out the coefficient of y to get the correct format:
  $(x - 3)^2 = -1(y - 15)$
• Identify the vertex:
  • Vertex: $(3, 15)$
• Determine the direction of opening:
  • Because of the negative sign, the parabola opens downward.
• Find the value of p:
  • The standard form is $4p(y - k) = (x - h)^2$
  • In this case, $4p = -1$
  • Solve for p: $p = -\frac{1}{4}$
• Calculate the focus:
  • Since the parabola opens downward, subtract $|p|$ from the y-coordinate of the vertex.
  • Focus: $(3, 15 - \frac{1}{4}) = (3, 14.75)$
• Determine the directrix:
  • Add $|p|$ to the y-coordinate of the vertex.
  • Directrix: $y = 15 + \frac{1}{4} = 15.25$