Vertex and Axis of Symmetry of a Parabola

Ask a question about a quadratic — where it turns, what its greatest value is, how to sketch it, where its mirror line runs — and the answer is almost always the vertex. It is the single most valuable point on the curve, and there are three good ways to find it.

The Vertex and the Axis

The vertex is the turning point of a parabola: the lowest point if the curve opens upward, the highest if it opens downward.

The axis of symmetry is the vertical line through the vertex that splits the curve into two mirror halves. Since it is vertical and passes through the vertex , its equation is always

Two details students routinely lose marks on:

• The axis is a line, so it needs an equation. Write , not just “2”.
• The vertex is a point, so it needs both coordinates. Write , not just .

Symmetry is what makes the axis practical. Any two points on the parabola at the same height are the same distance from it, so plotting one side of the curve gives you the other side free.

Method 1: The −b/2a Formula

For , the axis of symmetry is

Then substitute that back into the original equation to get the -coordinate. The formula gives you the axis; the substitution gives you the vertex.

For :

So the vertex is and the axis is .

Watch the signs. The formula already contains a minus, so a negative produces a positive answer. Writing explicitly before substituting prevents most errors here.

Where the Formula Comes From

It is not arbitrary. The two -intercepts of a parabola are mirror images of each other, so the axis must sit exactly halfway between them.

The quadratic formula gives those roots as

Average them. The two square-root terms are equal and opposite, so they cancel, leaving

Notice the discriminant disappeared. That is why the formula still works when the parabola has no real roots at all — the axis exists whether or not the curve ever crosses the -axis.

Method 2: Completing the Square

Completing the square converts standard form into vertex form, after which the vertex is simply readable.

Take :

Vertex , axis .

The step people get wrong is the bookkeeping. Here 9 was added inside the bracket, so 9 must be subtracted outside to keep the expression equal to what it was. Nothing is added to the other side — this is an expression being rewritten, not an equation being balanced, which is the difference from the circle version of the same technique.

When , factor it out of the -terms first:

Vertex .

Method 3: Average the Roots

If the -intercepts are easy to find, this is the quickest method of the three. The roots are symmetric, so:

For , factoring gives roots at and , so . Substituting back gives .

This only works when the parabola actually has two real x-intercepts. When it doesn’t, fall back on one of the other two methods.

Maximum or Minimum

The sign of settles it, and nothing else does:

• : opens upward, vertex is the minimum. The range is .
• : opens downward, vertex is the maximum. The range is .

Note which coordinate is the answer. “Where is the maximum?” wants . “What is the maximum?” wants . Applied questions are usually asking for , and quoting instead is a common way to lose the mark.

Worked Example A: Vertex From Standard Form

Find the vertex and axis of .

Vertex , axis . Since , it is a minimum.

Worked Example B: A Negative Leading Coefficient

Find the vertex of .

Vertex . Because , this is a maximum value of 4.

Worked Example C: By Completing the Square

Write in vertex form.

Half of 10 is 5, and :

Vertex .

Worked Example D: Maximum Height

A ball is thrown so that its height after seconds is . When does it peak, and how high does it get?

The vertex gives both answers: is when, is how high. The ball is back on the ground at , which the symmetry predicts — it takes as long to fall as it did to rise.

Worked Example E: Largest Area

A farmer has 40 m of fencing for three sides of a rectangular pen against a wall. What width gives the greatest area?

With width on the two perpendicular sides, the remaining side is , so

A width of 10 m gives area m². The negative confirms this is a maximum rather than a minimum.

Common Mistakes to Avoid

• Dropping the minus in the formula. It is , so gives .
• Stopping at the axis. is the axis; the vertex needs the too.
• Quoting when the question wants . “What is the maximum value” is asking for the -coordinate.
• Balancing both sides when completing the square on a function. Add and subtract on the same side — you are rewriting, not solving.
• Forgetting to factor out first. With the constant you add gets multiplied by on the way back out.
• Averaging the roots when there are none. That method needs two real intercepts.
• Writing the axis without ””. A line is an equation, not a number.

Where the Vertex Leads Next

• Optimisation. Every “largest area” or “maximum profit” question with a quadratic model is a vertex calculation.
• Range. The vertex fixes one end of the range: or .
• Transformations. In , and are exactly the horizontal and vertical shifts.
• Calculus. Setting the derivative to zero locates the same point; gives .
• Symmetry in other curves. Ellipses and hyperbolas have centres that play a similar organising role.

Practice Problems

Work each one before opening the answer.

Problem 1. Find the axis of symmetry of .

Problem 2. Find the vertex of .

Problem 3. Find the vertex of .

Problem 4. State the vertex of .

Problem 5. A parabola has -intercepts at and . What is its axis of symmetry?

Problem 6. Write in vertex form.

Problem 7. Does have a maximum or a minimum, and what is it?

Problem 8. Find the vertex of by completing the square.

Problem 9. What is the range of ?

Problem 10. A rocket’s height is . What is its greatest height?

Problem 11. Two points on a parabola, and , have the same height. Where is the axis?

Problem 12. A rectangle has perimeter 24 cm. What dimensions give the largest area?

Quick Reference

The vertex is the anchor of every parabola sketch, and locating it from standard form uses completing the square or the quadratic formula. It also fixes the range of the function. To drill the algebra, try the completing the square practice problems. More Algebra lessons are available.