Key takeaway: The vertex of a parabola is its turning point, the highest or lowest point on the curve, located at x = −b/(2a) for a quadratic in standard form.
Grade 9 – Algebra 1F-IF.C.8aFunctions and Graphs
Definition of Vertex of a Parabola
The vertex of a parabola is the point where the curve changes direction. On an upward-opening parabola it is the minimum point, and on a downward-opening one it is the maximum point. It lies on the axis of symmetry, exactly halfway between the two x-intercepts when they exist.
The vertex answers optimization questions: the greatest height of a thrown ball, the lowest cost of a production run, the price that gives the most profit. To find it from standard form, first compute the \(x\)-coordinate with a formula, then substitute that value back into the function to get the \(y\)-coordinate.
Formula and notation
The vertex of a parabola given by \(y = ax^{2} + bx + c\) is the point \((h, k)\) with
\[h = -\frac{b}{2a}, \qquad k = f(h)\]
In vertex form \(y = a(x – h)^{2} + k\), the vertex can be read directly: \((h, k)\).
Key facts
- The vertex is the minimum if a is positive and the maximum if a is negative.
- Its x-coordinate is the same as the line of symmetry.
- Find the y-coordinate by substituting the x-coordinate into the function.
- Shifting the vertex of a parabola translates the whole graph with it.
Vertex of a Parabola example
Find the vertex of \(f(x) = 2x^{2} – 12x + 7\) and say whether it is a minimum or a maximum.
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Identify \(a = 2\) and \(b = -12\). Then \(h = -\frac{-12}{2(2)} = \frac{12}{4} = 3\).
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Substitute: \(f(3) = 2(9) – 12(3) + 7 = 18 – 36 + 7 = -11\).
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Since \(a = 2 > 0\), the parabola opens upward.
The vertex is \((3, -11)\), a minimum.
Common mistake
Watch out
Lucia uses \(h = \frac{b}{2a}\) and gets \(\frac{-12}{4} = -3\), then reports the vertex as \((-3, 61)\).
Why it is wrong
The formula starts with a minus sign: \(h = -\frac{b}{2a}\). With \(b = -12\), the two negatives make a positive, giving \(h = 3\). Her point \((-3, 61)\) is on the graph but is far above the low point; the minimum value is \(-11\).
Related terms and lessons
- Parabola
- Axis of Symmetry
- Vertex Form
- Quadratic Function
- Standard Form of a Quadratic
- Completing the Square
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