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how to find axis of symmetry | axis definition,what does axis mean for glasses

  • Author: Profitaxis
  • Published On: January 28, 2025
  • Category:Latest News
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How to Find the Axis of Symmetry

1. The Basics of Parabolas

A parabola is a curve that can open upward, downward, left, or right, depending on its equation. It has a lowest (or highest) point called the vertex. The axis of symmetry passes through the vertex, splitting the parabola into two mirror-image halves.

2. Parabolas That Open Upward or Downward

The equation for these parabolas is typically written in the standard form:

y = ax² + bx + c

Vertex Formula: The x-coordinate of the vertex is given by:

x = -b / 2a

Once you find the x-coordinate, the axis of symmetry is a vertical line, represented by:

x = (x-coordinate of the vertex)

Example

Given the equation: y = 3x² - 12x + 7

Find the x-coordinate of the vertex:

x = -(-12) / (2 * 3) = 12 / 6 = 2

The axis of symmetry is: x = 2

Using a graphing tool confirms that the line x = 2 divides the parabola into two equal halves.

3. Parabolas That Open Left or Right

The equation for these parabolas is typically written in the form:

x = ay² + by + c

Vertex Formula: The y-coordinate of the vertex is found similarly:

y = -b / 2a

Here, the axis of symmetry is a horizontal line, represented by:

y = (y-coordinate of the vertex)

Example

Given the equation: x = 3(y - 2)² + 4

The vertex is at (4, 2). The axis of symmetry is:

y = 2

This parabola opens to the right because the coefficient a = 3 is positive.

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How to Find the Axis of Symmetry

1. The Basics of Parabolas

A parabola is a curve that can open upward, downward, left, or right, depending on its equation. It has a lowest (or highest) point called the vertex. The axis of symmetry passes through the vertex, splitting the parabola into two mirror-image halves.

2. Parabolas That Open Upward or Downward

The equation for these parabolas is typically written in the standard form:

y = ax² + bx + c

Vertex Formula: The x-coordinate of the vertex is given by:

x = -b / 2a

Once you find the x-coordinate, the axis of symmetry is a vertical line, represented by:

x = (x-coordinate of the vertex)

Example

Given the equation: y = 3x² - 12x + 7

Find the x-coordinate of the vertex:

x = -(-12) / (2 * 3) = 12 / 6 = 2

The axis of symmetry is: x = 2

Using a graphing tool confirms that the line x = 2 divides the parabola into two equal halves.

3. Parabolas That Open Left or Right

The equation for these parabolas is typically written in the form:

x = ay² + by + c

Vertex Formula: The y-coordinate of the vertex is found similarly:

y = -b / 2a

Here, the axis of symmetry is a horizontal line, represented by:

y = (y-coordinate of the vertex)

Example

Given the equation: x = 3(y - 2)² + 4

The vertex is at (4, 2). The axis of symmetry is:

y = 2

This parabola opens to the right because the coefficient a = 3 is positive.

4. Parabolas in Vertex Form

When a quadratic equation is written in vertex form, it becomes easier to identify the axis of symmetry.

y = a(x - h)² + k

Here:
(h, k) is the vertex of the parabola.
If the parabola opens upward or downward, the axis of symmetry is:

x = h

Example

Given the equation: y = 2(x - 3)² - 5

The vertex is (3, -5). The axis of symmetry is:

x = 3

5. Parabolas in Factored Form

When a quadratic equation is written in factored form, such as:

y = a(x - r₁)(x - r₂)

The axis of symmetry can be found by taking the average of the roots r₁ and r₂:

x = (r₁ + r₂) / 2

Example

Given the equation: y = (x - 1)(x - 5)

The roots are r₁ = 1 and r₂ = 5. The axis of symmetry is:

x = (1 + 5) / 2 = 6 / 2 = 3

6. Key Takeaways

  • For parabolas that open upward or downward: The axis of symmetry is vertical (x = constant).
  • For parabolas that open left or right: The axis of symmetry is horizontal (y = constant).
  • Use the vertex or the roots (if available) to determine the axis of symmetry.
  • Understanding the axis of symmetry not only helps in graphing parabolas but also in solving quadratic problems effectively.

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