The Axis of Symmetry
Part of the Quadratics & Complex Numbers section of Coddy's Algebra 2 journey. Lesson 1 of 51.
Every parabola is a mirror image of itself. The mirror is a vertical line called the axis of symmetry, and the vertex sits on it. In vertex form you read the axis straight off the bracket. In standard form, y = ax^2 + bx + c, there is no bracket to read, but there is a formula:
x = -b/(2a)
Here is why. The points of y = ax^2 + bx + c at height c are where ax^2 + bx = 0, that is x(ax + b) = 0. So x is 0 or -b/a. Two points at the same height are mirror images, so the axis runs exactly halfway between them, at half of -b/a, which is -b/(2a).
Take f(x) = 2x^2 + 8x + 1. Here a is 2 and b is 8, so the axis is x = -8/(2 × 2), which is -2. Notice that c never enters: sliding the parabola up or down does not move its mirror line.
Challenge
The parabola f(x) = 3x^2 - 12x + 7 has its axis of symmetry at the line x = h. Find h. On the blank board, write h = ... with a and b put into -b/(2a), then a second line with h worked out.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Quadratics & Complex Numbers
1The Vertex
The Axis of SymmetryThe Vertex PointMaximum or MinimumDrawing from Standard FormRecap: The Vertex2Completing the Square
Perfect Square TrinomialsThe Missing NumberCompleting the SquareIrrational AnswersWhen a Is Not 1Recap: Completing the Square5Complex Arithmetic
Adding and SubtractingMultiplyingThe ConjugateDividingA Pure Imaginary BottomRecap: Complex Arithmetic8Lines Meet Parabolas
Two CrossingsJust TouchingNo MeetingA Circle and a LineRecap: Lines and Parabolas