Axis of Symmetry

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Axis of Symmetry: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Axis of symmetry is the vertical mirror line x = −b/(2a) running through a parabola’s vertex, so the curve folds onto itself along it.

Algebra 1Functions and GraphsF-IF.C.7a

What is Axis of Symmetry?

Crease a sketch of a parabola straight down the middle and the two halves lie perfectly on top of each other; that crease is the axis of symmetry. It always goes through the vertex.

For \(y = ax^2 + bx + c\), the crease has equation \(x = -\frac{b}{2a}\). A point on one side has a twin on the other side, at identical height and equal distance from the crease, so the line sits halfway between any such twins, including the pair of x-intercepts.

How to spot it

  • A question asks for the line that cuts the curve into two matching pieces.
  • One point on a parabola is known and its twin at the same height is wanted.
  • Both x-intercepts are known, and the turning point’s x-value is requested.
  • A quadratic in standard form is given and the turning point must be located with no sketch.

The method

  1. Read a and b from \(ax^2 + bx + c\).
  2. Compute \(x = -\frac{b}{2a}\) and state it as a vertical line.
  3. To find a twin, measure how far the known point is from that line and step the same distance to the opposite side.
  4. Confirm that both points return the same height when substituted.
vertex (−1, −4)x = −1(−3, 0)(1, 0)(−2, −3)(0, −3)y = x² + 2x − 3
The dashed line x = −1 is halfway between mirror points such as (−3, 0) and (1, 0).

Axis of Symmetry: solved example

Problem

Given \(f(x) = 3x^2 – 18x + 4\), locate the mirror line and then the twin of the point \((0, 4)\).

Solution

With \(a = 3\) and \(b = -18\), the line is \(x = -\frac{-18}{2(3)} = 3\), so the answer is \(x = 3\).

The point \((0, 4)\) lies 3 units to the left of that line, so its twin is 3 units to the right: \((6, 4)\). Testing, \(f(6) = 108 – 108 + 4 = 4\). The twin is \((6, 4)\).

Try it

Question 1

State the mirror line of \(y = -2x^2 + 8x + 1\).

Show the answer

\(x = -\frac{8}{2(-2)} = 2\).

Question 2

A curve has mirror line \(x = -1\) and passes through \((4, 6)\). Name its twin point at height 6.

Show the answer

The point is 5 units right of the line, so the twin is 5 units left: \((-6, 6)\).

Axis of Symmetry vs. Vertex of a Parabola

The axis of symmetry is an entire line with an equation of the form \(x = h\). The turning point is one location on it, a pair \((h, k)\). If a question wants an equation, give the line; if it wants coordinates, substitute to find the height too.

Vertex of a Parabola

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