Algebra
Vertex of a Parabola – Finding the Turning Point
The vertex of a parabola is its single turning point – the lowest point if it opens upward, or the highest point if it opens downward. The vertex's x-coordinate is always given by the formula x = −b ÷ (2a); once you have that, substitute it back into the original equation to find the matching y-coordinate.
This formula isn't a coincidence – it comes directly from completing the square. Rewriting y = ax² + bx + c in vertex form reveals that the turning point sits exactly at x = −b/(2a), which is precisely why completing the square and finding a vertex are two sides of the very same idea.
Finding the Vertex
The vertex's x-coordinate is x = −b ÷ (2a). Substitute this x-value back into the equation to find the y-coordinate.
On the curve y = x² − 4x + 3, the vertex is the single turning point — the lowest point on this upward-opening curve, where it stops falling and starts rising.
For this curve, a = 1 and b = −4, so the vertex's x-coordinate is x = −(−4) ÷ (2 × 1) = 4 ÷ 2 = 2. Substituting back: y = 2² − 4(2) + 3 = 4 − 8 + 3 = −1, confirming the vertex is exactly where the picture shows it: (2, −1).
x = −(−8) ÷ (2 × 2) = 8 ÷ 4 = 2.
y = 2(2)² − 8(2) + 3 = 8 − 16 + 3 = −5. The vertex is (2, −5).
Real-Life Application
- Maximising profit: the vertex of a profit curve shows the price that maximises profit.
- Projectile peak: the vertex of a height-vs-time graph shows the maximum height reached.
- Minimising cost: the vertex of a cost curve shows the quantity that minimises cost.
Key Takeaways
- The vertex is a parabola's single turning point.
- Its x-coordinate is always x = −b ÷ (2a).
- Substitute the vertex's x-value into the equation to find its y-coordinate.