Formula for axis of symmetry of a parabola

📚 Understanding the Axis of Symmetry of a Parabola

The axis of symmetry is an imaginary vertical line that passes through the vertex of a parabola, dividing it into two perfectly symmetrical halves. Think of it like folding a parabola in half; the crease is the axis of symmetry.

📜 Historical Context

The study of parabolas dates back to ancient Greece, with mathematicians like Menaechmus exploring conic sections. Apollonius of Perga further developed the theory in his work "Conics." While the concept of an axis of symmetry wasn't explicitly defined as such in early works, the symmetrical nature of the parabola was recognized and utilized in geometric constructions and problem-solving.

🔑 Key Principles and the Formula

The most common way to find the axis of symmetry is by using the standard form of a quadratic equation:

$y = ax^2 + bx + c$

The formula for the axis of symmetry is:

$x = \frac{-b}{2a}$

Where 'a' and 'b' are the coefficients of the $x^2$ and $x$ terms, respectively.

• 🔢 Identify 'a' and 'b': Carefully note the values of 'a' and 'b' from your quadratic equation.
• ➖ Apply the Formula: Substitute 'a' and 'b' into the formula $x = \frac{-b}{2a}$.
• ➗ Simplify: Perform the calculation to find the x-value of the axis of symmetry. This x-value is the equation of the vertical line.

⚙️ Real-World Examples

Let's look at a few practical examples.

Example 1:

Consider the parabola defined by the equation: $y = 2x^2 + 8x - 5$

Here, $a = 2$ and $b = 8$. Applying the formula:

$x = \frac{-8}{2(2)} = \frac{-8}{4} = -2$

Therefore, the axis of symmetry is the vertical line $x = -2$.

Example 2:

Consider the parabola defined by the equation: $y = -x^2 + 4x + 1$

Here, $a = -1$ and $b = 4$. Applying the formula:

$x = \frac{-4}{2(-1)} = \frac{-4}{-2} = 2$

Therefore, the axis of symmetry is the vertical line $x = 2$.

✍️ Practice Quiz

Find the axis of symmetry for the following parabolas:

1. 🔍 $y = x^2 - 6x + 2$
2. 🧮 $y = 3x^2 + 12x - 10$
3. 📈 $y = -2x^2 - 8x + 5$

Answers:

1. 💡 $x = 3$
2. 🧪 $x = -2$
3. 📚 $x = -2$

🎯 Conclusion

Understanding the axis of symmetry is crucial for analyzing parabolas. The formula $x = \frac{-b}{2a}$ provides a straightforward method for finding it, making it easier to graph and analyze quadratic functions. With practice and these examples, you'll master this concept in no time!