Axis of Symmetry - Algebra-1

1. Fundamental Concepts

Definition: The axis of symmetry is a vertical line (perpendicular to the x-axis) about which the graph of a quadratic function (parabola) is symmetric. That is, the graph on both sides of the line is a mirror image of each other.
Expression: The equation of the axis of symmetry is usually written as , where h is the x-coordinate of the parabola's vertex.
Relationship with the Vertex: The axis of symmetry always passes through the vertex of the parabola. It is a direct reflection of the vertex's x-coordinate and the core symbol of the parabola's symmetry.

Uniqueness: Each parabola has exactly one axis of symmetry, and it is always perpendicular to the x-axis (i.e., parallel to the y-axis).
Manifestation of Symmetry: If a point lies on the parabola, its symmetric point with respect to the axis of symmetry must also lie on the parabola. The axis of symmetry is the perpendicular bisector of the line segment connecting these two points, satisfying (i.e., ).
Connection with Function Forms:
- In the vertex form , the axis of symmetry is directly ;
- In the general form , the formula for the axis of symmetry is (derived from the x-coordinate of the vertex).

1: Given the quadratic function , find its axis of symmetry.
Solution: From the vertex form, the x-coordinate of the vertex is , so the axis of symmetry is .
2: Given the quadratic function , find its axis of symmetry.
Solution: Method 1 (Formula Method): , . Substituting into the formula, we get , so the axis of symmetry is . Method 2 (Converting to Vertex Form): By completing the square, we obtain , so the axis of symmetry is .
3: A parabola passes through the points and , and one of its x-intercepts is . Find its axis of symmetry.
Solution: The points and have the same y-coordinate, so they are symmetric points on the parabola. Thus, the axis of symmetry is the average of their x-coordinates, i.e., . (Verification: The symmetric points satisfy , which conforms to the property of the axis of symmetry, so the solution is valid without relying on other intercepts.)

Methods to Find the Axis of Symmetry:
- Directly read from the vertex form: For a function in the form , the axis of symmetry is ;
- Calculate using the formula for the general form: Use (applicable to the general form );
- Derive from symmetric points: If two points and with the same y-coordinate on the parabola are known, the axis of symmetry is .
Using the Axis of Symmetry to Solve Problems:
- Find symmetric points: Given a point on the parabola and the axis of symmetry , its symmetric point can be found as ;
- Simplify calculations: When finding the x-intercepts, extrema, or properties of the parabola, the symmetry of the axis of symmetry can reduce the amount of calculation (for example, if one intercept is known, the other can be quickly found using symmetry).