Skill: Identify the vertex and axis of symmetry of a parabola

The vertex is the turning point of a parabola. It is the minimum point if the parabola opens upward and the maximum point if it opens downward.

The axis of symmetry is the vertical line that cuts the parabola into two matching halves.

For a quadratic in standard form,

$$y=ax^2+bx+c,$$

find the axis of symmetry using

$$x=-\frac{b}{2a}.$$

Then substitute that $x$-value into the equation to find the $y$-coordinate of the vertex.

Worked example

Find the vertex and axis of symmetry of

$$y=2x^2-8x+3.$$

Step 1: Identify $a$ and $b$.

Compare the equation with $y=ax^2+bx+c$:

• $a=2$
• $b=-8$

Step 2: Find the axis of symmetry.

$$x=-\frac{b}{2a}$$

Substitute $a=2$ and $b=-8$:

$$x=-\frac{-8}{2(2)}=\frac{8}{4}=2$$

So, the axis of symmetry is

$$\boxed{x=2}.$$

Step 3: Find the vertex’s $y$-coordinate.

Substitute $x=2$ into the original equation:

$$y=2(2)^2-8(2)+3$$ $$y=2(4)-16+3$$ $$y=8-16+3=-5$$

Therefore, the vertex is

$$\boxed{(2,-5)}.$$

The parabola’s turning point is $(2,-5)$, and the vertical line $x=2$ passes through it and divides the parabola into mirror-image halves.