Maximum/Minimum & the Vertex

Every quadratic has exactly one turning point, the vertex, and 0606 asks about it in three costumes: “state the minimum value”, “find the coordinates of the turning point”, “state the value of $x$ for which $f$ is least”. All three are the same question, answered from completed-square form.

Reading the vertex

From $f(x) = a(x + p)^2 + q$:

• Vertex: $(-p,\ q)$
• $a > 0$: parabola opens upward, so $q$ is the minimum value, at $x = -p$
• $a < 0$: opens downward, so $q$ is the maximum value, at $x = -p$

$f(x) = 2(x - 3)^2 + 5$: minimum value 5, at $x = 3$, vertex $(3,\ 5)$.

Watch the sign convention: $(x - 3)^2$ means the vertex is at $+3$. Reading $-p$ from $(x + p)^2$ is where the marks leak.

Answer the question that was asked

0606 distinguishes the value from the location:

• “State the minimum value of $f$” gives 5 (a $y$-value)
• “State the value of $x$ at which the minimum occurs” gives 3
• “Find the coordinates of the vertex” gives $(3,\ 5)$

Giving coordinates when a value was asked usually survives; giving $x = 3$ as “the minimum value” doesn’t. Read the command, answer its object.

Where it appears in harder questions

• Sketches: the labelled vertex is a standing B mark on quadratic and modulus-quadratic sketches.
• Range: the vertex value is the bound, range of a quadratic.
• One-one domains: “smallest $k$ such that $f$ is one-one for $x \ge k$” is the vertex $x$-coordinate, one-one functions.
• Applied max/min: “the height of the ball is $h(t) = -5t^2 + 20t + 1$; find the greatest height”, complete the square (or note this is where calculus stationary points offer a second route; for a plain quadratic, completing the square is faster and calculator-free).

Common mistakes

• Vertex $x$-coordinate sign-flipped
• Max and min confused when $a < 0$
• Value vs location vs coordinates mismatched to the question
• Vertex quoted as the range bound when a restricted domain excludes it (the domain trap)

Full topic context: Quadratic Functions notes · prerequisite: completing the square.