Key takeaway: Vertex form writes a quadratic as y = a(x − h)² + k, so the vertex (h, k), the direction and the width of the parabola can be read straight from the equation.
Grade 9 – Algebra 1A-SSE.B.3bFunctions and Graphs
Definition of Vertex Form
Vertex form is the version of a quadratic that shows its turning point. In \(y = a(x – h)^{2} + k\), the numbers \(h\) and \(k\) are the coordinates of the vertex, and \(a\) is the same stretch factor as in standard form. The graph is the basic parabola \(y = x^{2}\) moved right \(h\) units, up \(k\) units and stretched or flipped by \(a\).
Because the squared part is never negative, the sign of \(a\) tells whether \(k\) is the minimum (\(a > 0\)) or the maximum (\(a < 0\)). The form is especially helpful for sketching a graph quickly and for solving optimization problems. To use it for zeros, set \(y = 0\) and take square roots.
Formula and notation
The vertex form with vertex \((h, k)\) is
\[y = a(x – h)^{2} + k\]
Notice the minus sign inside the parentheses: \(y = (x + 4)^{2}\) has \(h = -4\), since \(x + 4 = x – (-4)\). The axis of symmetry is \(x = h\).
Key facts
- The vertex is (h, k); the h value has the opposite sign from what appears in the parentheses.
- If a is negative, the parabola opens downward.
- Larger absolute values of a make a narrower parabola.
- Expanding the square and combining terms converts vertex form to standard form.
Vertex Form example
For \(y = -\frac{1}{2}(x – 6)^{2} + 9\), find the vertex, the direction of opening, the \(y\)-intercept and the zeros.
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Compare with \(a(x – h)^{2} + k\): \(a = -\frac{1}{2}\), \(h = 6\), \(k = 9\). The vertex is \((6, 9)\) and it opens downward.
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Put \(x = 0\): \(-\frac{1}{2}(36) + 9 = -9\), so the \(y\)-intercept is \((0, -9)\).
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Put \(y = 0\): \((x – 6)^{2} = 18\), so \(x = 6 \pm 3\sqrt{2}\), about 1.76 and 10.24.
Vertex \((6, 9)\), opens down, \(y\)-intercept \(-9\), zeros \(6 \pm 3\sqrt{2}\).
Common mistake
Watch out
From \(y = (x + 4)^{2} – 13\), Tia states that the vertex is \((4, -13)\).
Why it is wrong
The pattern is \(x – h\). Since \(x + 4 = x – (-4)\), the value of \(h\) is \(-4\), so the vertex is \((-4, -13)\). Check by substituting \(x = -4\): the squared part is 0 and \(y = -13\), the lowest point.
Related terms and lessons
- Vertex of a Parabola
- Standard Form of a Quadratic
- Completing the Square
- Transformations of Functions
- Parabola
- Axis of Symmetry
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