Key takeaway: The axis of symmetry of a parabola is the vertical line x = −b/(2a) that passes through the vertex and divides the graph into two mirror-image halves.
Grade 9 – Algebra 1F-IF.C.7aFunctions and Graphs
Definition of Axis of Symmetry
The axis of symmetry is a mirror line for a parabola. If you fold the graph along it, every point on one side lands exactly on a point of the other side at the same height. The line always runs through the vertex, and its equation has the form \(x = h\).
Symmetry gives shortcuts. If you know one point on a parabola, its mirror image across the axis is also on the curve. If the parabola has x-intercepts, the axis lies halfway between them, so averaging the two zeros gives the axis. Equal heights at two different inputs also reveal the axis: it sits at their midpoint.
Formula and notation
For \(y = ax^{2} + bx + c\), the line is
\[x = -\frac{b}{2a}\]
A point \((x, y)\) has the mirror partner \((2h – x, y)\), where \(x = h\) is the axis. In vertex form \(y = a(x – h)^{2} + k\) the axis is \(x = h\).
Key facts
- The axis is always vertical for the parabolas of Algebra 1.
- It goes through the vertex, so the vertex x-coordinate is the axis.
- Points at the same height are the same distance from the axis.
- The midpoint of the zeros of a parabola lies on the axis.
Axis of Symmetry example
Find the axis of symmetry of \(y = 3x^{2} + 18x + 20\) and verify it using two points at the same height.
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With \(a = 3\) and \(b = 18\): \(x = -\frac{18}{2(3)} = -3\).
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Test \(x = -1\): \(3 – 18 + 20 = 5\). Test \(x = -5\): \(75 – 90 + 20 = 5\).
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The inputs \(-5\) and \(-1\) give the same output, and their midpoint is \(\frac{-5 + (-1)}{2} = -3\).
The axis of symmetry is the line \(x = -3\).
Common mistake
Watch out
Nell writes the axis of symmetry of \(y = 3x^{2} + 18x + 20\) as \(y = -3\).
Why it is wrong
An upright parabola is symmetric across a vertical line, and vertical lines are named by their x-value: \(x = -3\). A horizontal line \(y = -3\) would cut across the graph and does not mirror it. Write the axis with \(x\) on the left side.
Related terms and lessons
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