The axis of symmetry is the vertical line x = −b / 2a. The parabola is a mirror image across it.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The vertex
The vertex sits on that axis. Substitute the x value back to get its y.
Maximum or minimum
A positive a opens upward, so the vertex is the minimum. A negative a makes it the maximum.
A shortcut
The axis lies exactly halfway between the two roots, when they exist.
Finding the vertex
The axis of symmetry is x = −b/(2a), and the vertex sits on it. Substitute that x back into the function to get the y-coordinate. Two steps, and they work for every quadratic regardless of whether it factors.
Why that x-value
A parabola is symmetric, so its vertex lies exactly halfway between the roots. Averaging the two roots from the quadratic formula gives −b/(2a) — the ± terms cancel. The formula is the midpoint of the roots.
Maximum or minimum
The vertex is the minimum if a > 0 and the maximum if a < 0. This is why quadratics model optimisation problems: maximum profit, maximum height, minimum cost all sit at a vertex.
Using symmetry to graph quickly
Plot the vertex, then one point either side at equal distances — their y-values match. Three points and the symmetry give a good sketch fast, without building a long table of values.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Dev Locate a Vertex
Dev works with y = x² − 6x + 5.
- Step 1
Here a = 1 and b = −6, so −b / 2a is 6 / 2 = 3.
Step 4: Your Turn
Practice makes it stick.
The Axis
Problem 1 of 2
y = x² − 8x + 1. What is the axis of symmetry, as an x value?
Max or Min
Problem 2 of 2
y = −x² + 4x. Is the vertex a maximum? 1 for yes, 0 for no.
Find the Turning Point
1 of 8
y = x² − 4x + 1. Axis of symmetry, as an x value?
2 of 8
y = x² − 4x + 1. What is the y of the vertex?
3 of 8
y = x² + 6x. Axis of symmetry, as an x value?
4 of 8
Roots at 2 and 8. Where is the axis of symmetry?
5 of 8
a = 3. Is the vertex a minimum? 1 for yes, 0 for no.
6 of 8
y = x². Where is the vertex, as an x value?
7 of 8
Roots at −4 and 6. Where is the axis of symmetry?
8 of 8
Select every true statement about the vertex.
Step 5: Quick Check
Show what you know.
Question 1 of 1
y = x² − 10x + 2. Axis of symmetry, as an x value?
What You Learned
- The axis of symmetry is x = −b / 2a, and the vertex sits on it.
- A positive a gives a minimum; a negative a gives a maximum.
- The axis is halfway between the roots whenever they exist.