The turning point of a quadratic graph is the vertex — the single point where the parabola changes from decreasing to increasing (a minimum) or from increasing to decreasing (a maximum). Every quadratic has exactly one turning point, and its x-coordinate is always on the axis of symmetry of the graph.
What is the turning point and why does it matter?
The quadratic y = ax² + bx + c is a parabola:
- If a > 0, the parabola opens upward (U-shape) and the turning point is a minimum.
- If a < 0, the parabola opens downward (∩-shape) and the turning point is a maximum.
The turning point is important because it gives:
- The minimum or maximum value of y
- The axis of symmetry (x = the x-coordinate of the vertex)
- A key point for sketching the graph accurately
How do you find the turning point using the vertex formula?
For any quadratic y = ax² + bx + c, the x-coordinate of the turning point is:
x = −b / (2a)
Substitute this x-value back into the equation to find y.
Worked example: Find the turning point of y = 2x² − 8x + 5.
Here a = 2, b = −8, c = 5.
x = −(−8) / (2 × 2) = 8 / 4 = 2
y = 2(2)² − 8(2) + 5 = 2(4) − 16 + 5 = 8 − 16 + 5 = −3
Turning point: (2, −3). Since a = 2 > 0, this is a minimum.
Axis of symmetry: x = 2.
How do you find the turning point by completing the square?
Completing the square rewrites y = ax² + bx + c in the form y = a(x − h)² + k, where the turning point is directly visible as (h, k).
Worked example: Find the turning point of y = x² + 6x + 7.
- Group the x terms: y = (x² + 6x) + 7
- Complete the square: half of 6 is 3; 3² = 9. Write (x + 3)² − 9:
y = (x + 3)² − 9 + 7 - Simplify: y = (x + 3)² − 2
The minimum is at the point where (x + 3)² = 0, i.e. x = −3, giving y = −2.
Turning point: (−3, −2). It is a minimum because the coefficient of x² is positive.
Worked example (negative a): Find the turning point of y = −x² + 4x + 1.
- y = −(x² − 4x) + 1
- Half of −4 is −2; (−2)² = 4. Write (x − 2)² − 4:
y = −[(x − 2)² − 4] + 1 - Expand the negative: y = −(x − 2)² + 4 + 1 = −(x − 2)² + 5
Turning point: (2, 5). It is a maximum because a = −1 < 0.
What is the axis of symmetry and how do you use it?
The axis of symmetry is the vertical line x = h passing through the turning point. Because a parabola is symmetric about this line:
- Any two x-values that are the same distance from the axis of symmetry produce the same y-value.
- If you know the x-coordinates of two points with the same y-value (for example, the two roots), the axis of symmetry is exactly halfway between them.
Worked example: A quadratic has roots at x = 1 and x = 7. What is the axis of symmetry?
Axis of symmetry: x = (1 + 7) / 2 = 4
So the vertex has x-coordinate 4.
What information do you need to sketch a quadratic graph?
A complete sketch requires:
| Feature | How to find it |
|---|---|
| Shape | a > 0 → U-shape; a < 0 → ∩-shape |
| Turning point | Complete the square or use x = −b/(2a) |
| y-intercept | Substitute x = 0: y = c |
| x-intercepts (roots) | Solve ax² + bx + c = 0 (if real roots exist) |
| Axis of symmetry | x = x-coordinate of turning point |
Worked example sketch summary for y = x² − 4x + 3:
- a = 1 > 0 → U-shape (minimum)
- Vertex: x = 4/2 = 2; y = 4 − 8 + 3 = −1 → minimum at (2, −1)
- y-intercept: x = 0 → y = 3 → (0, 3)
- Roots: x² − 4x + 3 = (x − 1)(x − 3) = 0 → x = 1 and x = 3 → (1, 0) and (3, 0)
- Axis of symmetry: x = 2 (checks out: midpoint of 1 and 3 is 2)
Frequently asked questions
Which method is better — the vertex formula or completing the square?
Both give the same answer. The vertex formula x = −b/(2a) is quicker for finding the coordinates only. Completing the square rewrites the whole equation in vertex form, which is more useful if you also need to sketch the graph, state the axis of symmetry, or answer questions about the minimum/maximum value.
Can the turning point be at a non-integer coordinate?
Yes. For example, y = x² + x + 1 has x = −1/2 at the turning point and y = 3/4. Examiners often choose such values deliberately. Work with fractions rather than rounding them.
How does the turning point relate to the discriminant?
If the discriminant b² − 4ac > 0, the parabola crosses the x-axis at two points and the vertex lies either above or below the axis depending on the sign of a. If b² − 4ac = 0, the vertex sits exactly on the x-axis (a repeated root). If b² − 4ac < 0, the parabola does not cross the x-axis and the vertex is above (minimum, a > 0) or below (maximum, a < 0) the axis.
What does the turning point tell us in a real-life context?
If a quadratic models a situation — for example, the height of a projectile over time — the turning point gives the maximum height (for ∩-shaped) and the time at which it occurs, or the minimum cost, profit threshold, etc. Always interpret the coordinates in the context of the problem.
For Socratic GCSE quadratic graphs practice with Professor Pi, see aitutors.me.