Steps to transform general form parabola equations using completing the square

📚 Understanding the General Form of a Parabola

The general form of a parabola equation is given by $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. This form is useful for understanding the basic structure of a parabola, but it doesn't immediately reveal key features like the vertex or axis of symmetry. Transforming this equation into vertex form makes these features much clearer.

📜 Historical Context and Significance

The study of parabolas dates back to ancient Greece, with mathematicians like Menaechmus exploring their properties. Parabolas gained further importance with the development of analytic geometry by René Descartes, allowing them to be described algebraically. The technique of completing the square has been used for centuries to solve quadratic equations and transform algebraic expressions. Its application to parabolas provides a powerful tool for analyzing their geometric properties.

🔑 Key Principles: Completing the Square

Completing the square involves rewriting a quadratic expression in the form $(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. Here's a step-by-step breakdown:

• 📝 Step 1: Factor out 'a': If $a \neq 1$, factor 'a' out of the $ax^2 + bx$ terms: $y = a(x^2 + \frac{b}{a}x) + c$.
• ➕ Step 2: Complete the Square: Take half of the coefficient of the $x$ term (i.e., $\frac{b}{2a}$), square it (i.e., $(\frac{b}{2a})^2$), and add and subtract it inside the parentheses: $y = a(x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 - (\frac{b}{2a})^2) + c$.
• ✍️ Step 3: Rewrite as a Square: Rewrite the expression inside the parentheses as a perfect square: $y = a((x + \frac{b}{2a})^2 - (\frac{b}{2a})^2) + c$.
• ➗ Step 4: Distribute and Simplify: Distribute the 'a' and simplify the expression: $y = a(x + \frac{b}{2a})^2 - a(\frac{b}{2a})^2 + c$.
• 📍 Step 5: Vertex Form: The equation is now in vertex form: $y = a(x - h)^2 + k$, where $h = -\frac{b}{2a}$ and $k = c - a(\frac{b}{2a})^2$.

💡 Example 1: Transforming $y = x^2 + 6x + 5$

Let's transform the equation $y = x^2 + 6x + 5$ into vertex form:

• 🔍 Step 1: $a = 1$, so no factoring is needed.
• ➕ Step 2: Half of 6 is 3, and $3^2 = 9$. Add and subtract 9: $y = x^2 + 6x + 9 - 9 + 5$.
• ✍️ Step 3: Rewrite as a square: $y = (x + 3)^2 - 9 + 5$.
• ➗ Step 4: Simplify: $y = (x + 3)^2 - 4$.
• 📍 Step 5: Vertex form: $y = (x - (-3))^2 + (-4)$. The vertex is $(-3, -4)$.

🧪 Example 2: Transforming $y = 2x^2 - 8x + 10$

Now, let's transform $y = 2x^2 - 8x + 10$:

• 📝 Step 1: Factor out 2: $y = 2(x^2 - 4x) + 10$.
• ➕ Step 2: Half of -4 is -2, and $(-2)^2 = 4$. Add and subtract 4 inside the parentheses: $y = 2(x^2 - 4x + 4 - 4) + 10$.
• ✍️ Step 3: Rewrite as a square: $y = 2((x - 2)^2 - 4) + 10$.
• ➗ Step 4: Distribute and simplify: $y = 2(x - 2)^2 - 8 + 10$.
• 📍 Step 5: Vertex form: $y = 2(x - 2)^2 + 2$. The vertex is $(2, 2)$.

🌍 Real-World Applications

Understanding parabolas and their vertex form has numerous real-world applications:

• 🛰️ Satellite Dishes: The shape of a satellite dish is parabolic, focusing signals at a single point.
• 🌉 Bridge Design: Parabolic arches are used in bridge construction for their structural strength.
• 🏀 Projectile Motion: The path of a projectile, like a ball thrown in the air, follows a parabolic trajectory.

📝 Conclusion

Transforming general form parabola equations using completing the square is a valuable skill in mathematics. By following the step-by-step process, you can easily convert equations into vertex form, revealing key features and enabling a deeper understanding of parabolas and their applications.