Quadratic Functions • Topic 2 of 5

Vertex Form

Vertex form, y = a(x − h)² + k, shows the vertex (h, k) directly and is ideal for graphing and finding maxima or minima. Note the sign: x − h means the vertex x-coordinate is +h, so y = (x + 3)² + 1 has vertex (−3, 1). From standard form you reach vertex form by completing the square, or find the vertex x with −b/(2a) and substitute to get k. The value of a is unchanged between forms. The SAT often gives or asks for vertex form because it makes the turning point and the transformation from y = x² obvious.

✅ Solved examples

1. What is the vertex of y = (x − 4)² + 2?
Vertex (h, k) = (4, 2).
2. What is the vertex of y = (x + 3)² − 5?
x + 3 = x − (−3), so the vertex is (−3, −5).
3. Find the vertex of y = x² − 6x + 5.
x = −b/(2a) = 3; y = 9 − 18 + 5 = −4, so (3, −4).
4. Write y = x² + 2x + 1 in vertex form.
x² + 2x + 1 = (x + 1)², so y = (x + 1)² + 0; vertex (−1, 0).

✏️ Practice — try these, take hints as needed

1. What is the vertex of y = (x − 2)² + 7?
Vertex is (h, k).
h = 2, k = 7.
—
(2, 7).
2. What is the vertex of y = (x + 5)² − 3?
x + 5 = x − (−5).
h = −5.
—
(−5, −3).
3. Find the vertex of y = x² − 8x + 10.
x = −b/(2a) = 4.
y = 16 − 32 + 10.
—
(4, −6).
4. What is the vertex of y = −(x − 1)² + 4?
h = 1, k = 4.
The leading − does not move it.
—
(1, 4).
5. Write y = x² − 4x + 4 in vertex form.
It is a perfect square.
(x − 2)².
—
(x − 2)² (vertex (2, 0)).

📝 Topic test — 8 questions

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