Matemáticas

What Is a Turning Point on a Graph?

Where a graph stops rising and starts falling, or the reverse. The minimum of y = x² − 6x + 5 read off completed-square form, plus max, min and stationary.

Turning point

A point where a graph stops rising and begins to fall, or stops falling and begins to rise.

También llamado: Vertex, Maximum or minimum point
Dónde lo encuentra el alumnado: Grade 9 or 10 mathematics, once quadratic graphs are sketched and completing the square is available to locate the point exactly.

La respuesta corta
A turning point is where a curve changes direction — from rising to falling, which is a maximum, or from falling to rising, which is a minimum. Completing the square locates it exactly: y = x² − 6x + 5 becomes y = (x − 3)² − 4, so the minimum sits at (3, −4).

Un ejemplo

y = x² − 6x + 5 = (x − 3)² − 4 → minimum at (3, −4)

A squared bracket can never be negative, so (x − 3)² is at its smallest when the bracket is zero, which happens at x = 3. At that point the whole expression is 0 − 4 = −4, and every other value of x makes it larger. That is an argument rather than a rule, and it is why completed-square form gives the coordinates directly: the number inside the bracket locates the point across, the number outside locates it up or down.

Completing the square hands you the coordinates

For a quadratic there is no searching involved. Rewrite it as a squared bracket plus a constant and both coordinates are sitting in front of you — x = 3 from inside the bracket, y = −4 from the constant outside.

The sign inside the bracket catches people out every time. (x − 3)² gives a turning point at x = 3, while (x + 3)² gives one at x = −3. The bracket is zero where the point is, so solve x − 3 = 0 rather than reading the number as it appears.

There is a quicker route if you only need the x coordinate: it is always −b/2a, which for x² − 6x + 5 gives 6/2 = 3. Substituting that back into the original equation supplies the y coordinate. Completing the square gives both at once, which is why papers ask for it.

Maximum, minimum, and the one that is neither

The sign of the x² coefficient decides which kind you have, and it decides it before any working. Positive means the parabola opens upwards and the turning point is the lowest place on the curve, a minimum. Negative means it opens downwards and the point is a maximum. There is exactly one either way — a quadratic cannot have two.

Stationary point is the wider term, and the two words are not interchangeable. Every turning point is stationary, because the curve is momentarily flat there. But a curve can flatten out and then carry on in the same direction without turning, which is what y = x³ does at the origin. That is a stationary point and not a turning point, and it is why examiners sometimes ask you to justify which kind you have found rather than just name it.

What the point is actually used for

It answers the questions that ask for a best value. The minimum of a cost function is the cheapest option; the maximum of a height function is how high the ball went. In each case the y coordinate is the value asked for and the x coordinate is the condition that produces it, and confusing the two is a common way to answer the wrong question.

It also fixes the symmetry of the curve. A parabola is symmetrical about the vertical line through its turning point, so (3, −4) tells you the axis of symmetry is x = 3, and that the roots at 1 and 5 sit an equal distance either side of it. Knowing the turning point and one root gives you the other for free.

Preguntas frecuentes

How do you find the turning point of a quadratic?

Complete the square and read both coordinates off, or use x = −b/2a for the horizontal position and substitute it back for the vertical one. Completing the square is usually asked for because it produces the coordinates and the minimum value together, which later parts of the question tend to need.

Is the turning point the same as the vertex?

For a parabola, yes. Vertex is the standard word in American textbooks and on the SAT, turning point is the British phrasing, and both name the point where the curve changes direction. Vertex form and completed-square form are similarly two names for the same rewriting.

Can a graph have more than one turning point?

A quadratic has exactly one. A cubic has either two — one maximum and one minimum — or none at all, which is why cubic sketches vary so much in shape. The number possible rises with the degree of the equation, and the highest power is a good first guide to what to expect.

How do you tell a maximum from a minimum without a graph?

Look at the coefficient of x². Positive and the curve opens upwards, so the turning point is a minimum; negative and it opens downwards, so it is a maximum. For y = 5 − 2x² that coefficient is −2, so the point at (0, 5) is a maximum even though the equation begins with a positive number.

Última actualización 21 de agosto de 2026