Math101 · Independent Ontario learning library
Math101Axis of Symmetry
An axis of symmetry is a line that divides a figure or graph into mirror-image halves. For a vertical parabola $y=ax^2+bx+c$ with $a\ne0$, its axis is $x=-\frac{b}{2a}$.
The axis organizes a parabola’s symmetry, locates its turning point, and simplifies graphing and optimization. It connects standard, vertex, and factored forms.
Intuition and core definition
An axis of symmetry is a line that divides a figure or graph into mirror-image halves. For a vertical parabola $y=ax^2+bx+c$ with $a\ne0$, its axis is $x=-\frac{b}{2a}$. This vertical line passes through the vertex and pairs inputs equally spaced from it with equal outputs.
Notation, language, and conditions
In vertex form $y=a(x-h)^2+k$, the axis is immediately $x=h$. In factored form $y=a(x-r_1)(x-r_2)$ with two real roots, the axis lies halfway between them: $x=(r_1+r_2)/2$. The axis is an equation of a line, not merely the number $h$.
Why this idea matters
The axis of symmetry divides a parabola into mirror halves and identifies the input at which its vertex and paired outputs are organized.
A dependable method
- Identify the parabola’s form and coefficients accurately.
- Use $x=h$ in vertex form, $x=-b/(2a)$ in standard form, or the midpoint of two roots.
- Compute the corresponding $y$-value if the vertex is also requested.
- Reflect one point across the proposed line and compare outputs.
- Write the axis as $x=$ a number and check it passes through the vertex.
