Vertex form and standard form

Three faces of one parabola

Every quadratic can be written in different forms. Each shows different features directly.

The value of a is the same in all three: it controls the direction of opening and the width.

Reading vertex form

y = 2(x − 3)² + 5

• Vertex: (3, 5)
• Axis: x = 3
• Opens up (a = 2 > 0), so 5 is the minimum
• Narrower than y = x²

Watch the sign: y = (x + 4)² − 1 has vertex (−4, −1).

Vertex form → standard form

Expand and simplify.

y = 2(x − 3)² + 5

• = 2(x² − 6x + 9) + 5
• = 2x² − 12x + 18 + 5
• = 2x² − 12x + 23

Standard form → vertex form

Method 1: completing the square

y = x² + 6x + 1 = (x² + 6x + 9) − 9 + 1 = (x + 3)² − 8

Method 2: use h = −b/(2a)

y = 3x² − 12x + 7

• h = 12/6 = 2
• k = 3(4) − 24 + 7 = −5
• y = 3(x − 2)² − 5

Getting the vertex from factored form

The vertex is halfway between the zeros.

y = (x − 1)(x − 7): zeros 1 and 7 → h = 4 → k = (3)(−3) = −9 → vertex (4, −9).

Writing an equation from a vertex and a point

Example: Vertex (2, −3), passes through (4, 5).

• y = a(x − 2)² − 3
• 5 = a(2)² − 3 → 4a = 8 → a = 2
• y = 2(x − 2)² − 3

Which form to use?

Worked examples

Example 1: Vertex of y = −(x − 5)² + 2? (5, 2), a maximum.

Example 2: Convert y = x² − 8x + 10 to vertex form → (x − 4)² − 6.

Example 3: Expand y = 3(x + 1)² − 4 → 3x² + 6x − 1.

Common mistakes

• Sign of h: (x − h) means h is positive; (x + 4) means h = −4.
• Forgetting to multiply by a when expanding.
• Errors when a ≠ 1 while completing the square.

Try it yourself

1. Vertex and axis of y = 4(x + 2)² + 1?
2. Expand y = (x − 3)² − 2.
3. Write y = x² + 10x + 20 in vertex form.
4. Vertex of y = 2x² − 8x + 3?
5. Write the equation with vertex (−1, 4) through (1, −4).

Answers: 1) (−2, 1); x = −2 2) x² − 6x + 7 3) (x + 5)² − 5 4) (2, −5) 5) y = −2(x + 1)² + 4

Quick recap

• Standard form shows c; vertex form shows (h, k); factored form shows the zeros.
• Convert with expanding or completing the square.
• a is the same in every form.