How do you write the equation of the line of symmetry?
The equation of the line of symmetry for a parabola is typically a vertical line of the form x = constant, which perfectly divides the parabola into two mirrored halves. Its formula depends on the specific form of the quadratic equation.
Understanding the Line of Symmetry
For any quadratic function, which graphs as a parabola, the line of symmetry is a crucial element. It's a vertical line that passes through the vertex of the parabola, ensuring that every point on one side of the line has a corresponding point on the opposite side, equidistant from the line.
Writing the Equation for Standard Form
When a quadratic equation is presented in its standard form, y = ax² + bx + c, where a, b, and c are real numbers and a is not zero, the equation of the line of symmetry can be precisely determined using a straightforward formula.
Formula:
The line of symmetry is given by:
x = -b / 2a
Here's how to apply it:
- Identify 'a' and 'b': From your quadratic equation in standard form, pick out the coefficients a (the coefficient of x²) and b (the coefficient of x).
- Substitute Values: Plug these values into the formula x = -b / 2a.
- Calculate: Perform the calculation to find the x-coordinate, which is the equation of your vertical line of symmetry.
Example 1: Standard Form
Consider the quadratic equation y = 2x² + 8x + 3.
- Identify a and b:
- a = 2
- b = 8
- Apply the formula:
- x = -8 / (2 * 2)
- x = -8 / 4
- x = -2
Therefore, the equation of the line of symmetry for y = 2x² + 8x + 3 is x = -2.
Writing the Equation for Vertex Form
The vertex form of a quadratic equation, y = a(x - h)² + k, provides the line of symmetry directly within its structure. In this form, the point (h, k) represents the vertex of the parabola. Since the line of symmetry always passes through the vertex and is a vertical line, its equation is simply the x-coordinate of the vertex.
Formula:
The line of symmetry is given by:
x = h
Here's how to apply it:
- Identify 'h': From your quadratic equation in vertex form, identify the value of h. Remember that in the formula, it's (x - h), so if you have (x + 3)², then h would be -3.
- State the Equation: The line of symmetry is simply x = h.
Example 2: Vertex Form
Consider the quadratic equation y = -3(x - 4)² + 7.
- Identify h:
- The term is (x - 4)², so h = 4.
- State the equation:
- x = 4
Therefore, the equation of the line of symmetry for y = -3(x - 4)² + 7 is x = 4.
Summary Table
| Quadratic Form | Equation | Formula for Line of Symmetry |
|---|---|---|
| Standard Form | y = ax² + bx + c |
x = -b / 2a |
| Vertex Form | y = a(x - h)² + k |
x = h |
By understanding these two common forms of quadratic equations, you can easily determine and write the equation for the line of symmetry, a fundamental characteristic of parabolas.
