# How do you find the axis of symmetry if only given points (0,2) and (6,2)?

Sep 30, 2015

Axis of symmetry: $x = 3$

#### Explanation:

Assuming we are talking about a parabola in standard position (with either a vertical or horizontal axis of symmetry:

Since $y = 2$ has $2$ solutions for $x$
the axis of symmetry can not be horizontal.
(A horizontal axis of symmetry has single $x$ values for any single value of $y$).

So (based on the assumption (above) the axis of symmetry is vertical
i.e. the axis of symmetry is of the form $x = a$ for some constant $a$

and by definition of the axis of symmetry:
$\left(0 , 2\right)$ and $\left(6 , 2\right)$ must be reflections of each other in this axis
i.e. the $x$ values $0$ and $6$ must be equidistant from the value $a$
$\Rightarrow a = 3$
and the axis of symmetry is $x = 3$

Note that it is possible to have a parabola which has an axis of symmetry that is at an angle to both the X and Y-axis and which passes through the given points, but its axis of symmetry is impossible to determine from the given data.