Axis of Symmetry Calculator

Understanding the axis of symmetry

The axis of symmetry is a fundamental concept in algebra and geometry. For parabolas, it's a vertical line that divides the curve into two identical halves. Every point on one side of this line has a mirror point on the other side at the same distance from the axis.

Key concepts

• Axis of symmetry formula: For a parabola in standard form y = ax² + bx + c, the axis of symmetry is always x = -b/(2a). This vertical line passes through the vertex of the parabola.
• Vertex coordinates: The vertex is the point where the parabola reaches its maximum or minimum value. Its x-coordinate is -b/(2a), and the y-coordinate is found by substituting this x value back into the original equation.
• Vertex form: The equation y = a(x - h)² + k reveals the vertex (h, k) directly. Converting from standard form to vertex form makes it easier to identify the parabola's key features.
• Parabola direction: If a > 0, the parabola opens upward and has a minimum point at the vertex. If a < 0, it opens downward with a maximum point at the vertex.
• Other conic sections: Ellipses have two axes of symmetry (major and minor axes), while hyperbolas also have two perpendicular axes of symmetry through their center.

How to use this calculator

1. Select the shape type (parabola, ellipse, or hyperbola)
2. For parabolas, enter the coefficients a, b, and c from the standard form y = ax² + bx + c
3. Click "Calculate" to see the axis of symmetry, vertex coordinates, and vertex form
4. Review the step-by-step solution to understand the calculation process

Example calculations

Example 1: For the parabola y = 2x² + 8x + 5:

• a = 2, b = 8, c = 5
• Axis of symmetry: x = -8/(2×2) = -8/4 = -2
• Vertex x-coordinate: -2
• Vertex y-coordinate: 2(-2)² + 8(-2) + 5 = 8 - 16 + 5 = -3
• Vertex: (-2, -3)
• Vertex form: y = 2(x + 2)² - 3

Example 2: For the parabola y = -x² + 6x - 5:

• a = -1, b = 6, c = -5
• Axis of symmetry: x = -6/(2×(-1)) = -6/(-2) = 3
• Vertex x-coordinate: 3
• Vertex y-coordinate: -(3)² + 6(3) - 5 = -9 + 18 - 5 = 4
• Vertex: (3, 4)
• Vertex form: y = -(x - 3)² + 4

Frequently Asked Questions

What is the axis of symmetry?

The axis of symmetry is a vertical line that divides a parabola into two mirror images. For a parabola in the form y = ax² + bx + c, the axis of symmetry is x = -b/(2a). This line passes through the vertex of the parabola.

How do you find the axis of symmetry?

For a quadratic function y = ax² + bx + c, use the formula x = -b/(2a). The coefficients a and b come from the standard form of the equation. For example, in y = 2x² + 4x + 1, a = 2 and b = 4, so the axis of symmetry is x = -4/(2×2) = -1.

What is vertex form?

Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. The axis of symmetry is x = h. Converting from standard form to vertex form helps identify the parabola's maximum or minimum point and its line of symmetry.

Do ellipses have an axis of symmetry?

Yes, ellipses have two axes of symmetry: the major axis (longest diameter) and the minor axis (shortest diameter). These axes are perpendicular to each other and intersect at the center of the ellipse.

Can the axis of symmetry be horizontal?

For standard parabolas of the form y = ax² + bx + c, the axis of symmetry is always vertical. However, for parabolas that open horizontally (of the form x = ay² + by + c), the axis of symmetry would be horizontal.