Vertex form, standard form and factored form of a quadratic function: what each form shows at a glance (the vertex, the y-intercept, the roots), how to convert between them by completing the square, expanding and factoring, the vertex formula x = −b/2a, the axis of symmetry, the discriminant and the number of roots, worked examples and a printable sheet

A quadratic function is a polynomial of degree 2, and its graph is a parabola. The same parabola can be written three ways, and each way shows one feature with no work: the standard form shows the y-intercept, the vertex form shows the vertex, the factored form shows the roots. "Vertex form" alone is searched about 59,000 times a month in the United States, mostly by Algebra 1 and 2 students told to "write in vertex form" and unsure what the form is or how to reach it. The three forms, the conversions and the vertex formula are below with the printable.

Download the quadratic forms sheet (PDF) (the three forms, the conversions and the vertex formula on page 1, the worked examples and a practice set with answers on page 2; two pages, free).

The three forms

In every form a is the same number: it is the coefficient of x² once the brackets are expanded, and it controls the shape. |a| > 1 makes the parabola narrower than y = x², |a| < 1 wider, and a negative a flips it to open downward.

Reading each form: one example three ways

y = 2x² − 8x + 6 (standard): y-intercept 6, opens up. y = 2(x − 2)² − 2 (vertex): vertex (2, −2), minimum value −2, axis x = 2. y = 2(x − 1)(x − 3) (factored): roots 1 and 3.

Check: the vertex x = 2 is halfway between the roots 1 and 3; expanding the factored form gives 2(x² − 4x + 3) = 2x² − 8x + 6. One parabola.

Converting between the forms

Standard → vertex (completing the square). 1. Factor a out of the first two terms: y = a(x² + (b/a)x) + c. 2. Take half the x coefficient inside the bracket, square it, add and subtract it inside. 3. The first three terms inside are a perfect square; write them as (x + half)². Multiply the subtracted number by a and move it out.

y = 3x² + 12x + 5 = 3(x² + 4x) + 5 = 3(x² + 4x + 4 − 4) + 5 = 3(x + 2)² − 12 + 5 = 3(x + 2)² − 7. Vertex (−2, −7).

Standard → vertex (the vertex formula, faster). h = −b/(2a), k = f(h). For 3x² + 12x + 5: h = −12/6 = −2, k = 3(4) − 24 + 5 = −7. Same answer, and the way to go when the question only wants the vertex.

Vertex → standard. Expand the square and collect: 3(x + 2)² − 7 = 3(x² + 4x + 4) − 7 = 3x² + 12x + 12 − 7 = 3x² + 12x + 5.

Standard → factored. Factor out a if the coefficients share it, then factor the trinomial; if it will not factor, use the quadratic formula to find p and q and write a(x − p)(x − q). 2x² − 8x + 6 = 2(x² − 4x + 3) = 2(x − 1)(x − 3).

Factored → standard. Expand the two brackets (FOIL), then multiply by a.

Factored → vertex. The vertex is halfway between the roots: h = (p + q)/2, k = f(h). For 2(x − 1)(x − 3): h = 2, k = 2(1)(−1) = −2, so 2(x − 2)² − 2.

The discriminant

For ax² + bx + c = 0, the discriminant is D = b² − 4ac.

D a perfect square (0, 1, 4, 9, ...) means the quadratic factors over the integers; otherwise the roots are irrational and the quadratic formula is the route.

Worked examples

1. Write y = x² + 6x + 2 in vertex form. (x² + 6x + 9) − 9 + 2 = (x + 3)² − 7; vertex (−3, −7).
2. Find the vertex of y = −2x² + 8x − 3. h = −8/(−4) = 2; k = −8 + 16 − 3 = 5; (2, 5), a maximum because a < 0.
3. Write y = (x − 4)² − 9 in factored form. Difference of squares: (x − 4 − 3)(x − 4 + 3) = (x − 7)(x − 1); roots 7 and 1.
4. y-intercept of y = 2(x − 1)(x + 5). x = 0: 2(−1)(5) = −10.
5. How many roots has 3x² − 2x + 1 = 0? D = 4 − 12 = −8 < 0: none (real); the parabola opens up and sits above the axis.
6. Axis of symmetry of y = 5(x + 1.5)² − 2. x = −1.5.
7. Vertex form with a = −1 and vertex (3, 4), through (5, 0)? y = −(x − 3)² + 4; check x = 5: −4 + 4 = 0. Yes.
8. The maximum height of h(t) = −16t² + 64t + 5. t = −64/(−32) = 2; h = −64 + 128 + 5 = 69 at t = 2.

The mistakes that lose marks

• The sign of h: y = (x − 3)² has its vertex at x = +3. The bracket shows x minus h.
• Forgetting to multiply the subtracted square by a when completing the square with a ≠ 1 (the −4 inside 3(x² + 4x + 4 − 4) is −12 outside).
• Dropping a when converting to factored form: 2x² − 8x + 6 is 2(x − 1)(x − 3), not (x − 1)(x − 3).
• Reading the vertex off standard form as (b, c): it is at x = −b/2a, and y must be computed.
• Treating a negative discriminant as "no solutions at all": there are no real roots; the complex roots exist and Algebra 2 asks for them.

Where to get help

If the method is not sticking from a page, a 1-on-1 online class fixes it faster than another worksheet: one tutor with a master's degree, one child, on a shared whiteboard, in your time zone (7 am before school, after 4 pm and weekend mornings, US time; after-school slots for the UK, Canada, Australia and the UAE), for about US$10 to $13 an hour. The first 30-minute class is free: how 1-on-1 online tutoring works.

Related pages

• Graphing linear equations, four methods
• Exponential growth and decay
• The square root function and its graph

Questions parents ask

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