Vertex form calculator: y = a(x − h)² + k

Vertex form writes a quadratic as y = a(x − h)² + k, where (h, k) is the vertex: the lowest point of the parabola if a is positive, the highest if a is negative. From y = ax² + bx + c, the vertex is at h = −b ÷ 2a, and k is the value of y there. So y = 2x² − 12x + 13 becomes y = 2(x − 3)² − 5, with vertex (3, −5). With this vertex form calculator method, type a, b and c into our Quadratic Graph calculator to find and see the vertex, then write the vertex form.

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Quadratic Graph

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How to find vertex form with the calculator

1. Open the Quadratic Graph calculator and type a, b and c in the Equation section. For 2x² − 12x + 13, type 2, −12 and 13.
2. Under Analysis, choose Min / Max. It shows the vertex as a local minimum or maximum, such as f(3) = −5.
3. Write the vertex form: keep the same a, and put the vertex's x as h and its y as k: y = a(x − h)² + k.
4. To check, type your vertex form into the Custom Graph, such as 2(x−3)^2 − 5, and see that it draws the same parabola.

The calculator finds the vertex as a decimal, and looks for it between x = −20 and 20. It doesn't write out the completing-the-square working, so the method below shows how to get the exact vertex form by hand.

The vertex formula

h is also the axis of symmetry, x = h. Instead of the formula for k, you can put h back into the equation: k = ah² + bh + c.

Standard form to vertex form by completing the square

1. Take out a from the x² and x terms.
2. Inside the bracket, halve the x coefficient and square it: add it and subtract it.
3. Write the first three terms as a perfect square.
4. Multiply out the leftover number and tidy the constant.

Example 1: y = 2x² − 12x + 13

So y = 2(x − 3)² − 5, with vertex (3, −5). The formula agrees: h = 12 ÷ 4 = 3, and k = 13 − 144 ÷ 8 = −5. In the calculator, Min / Max shows f(3) = −5.

The parabola is symmetric about x = 3, and its lowest point is the vertex.

Example 2: when a is 1

For y = x² + 2x − 1, the calculator's opening equation, half of 2 is 1: y = (x + 1)² − 1 − 1 = (x + 1)² − 2. The vertex is (−1, −2). Note the sign: x + 1 means h = −1.

Example 3: a negative a

For y = −x² + 4x + 1, take out −1: y = −(x² − 4x) + 1 = −[(x − 2)² − 4] + 1 = −(x − 2)² + 5. The parabola opens downwards, so the vertex (2, 5) is a maximum, and the calculator labels it a local maximum.

Example 4: vertex form back to standard form

Expand the bracket. y = 3(x + 1)² − 5 = 3(x² + 2x + 1) − 5 = 3x² + 6x − 2. Type 3, 6 and −2 into the calculator and Min / Max gives the vertex (−1, −5), which matches.

For more on the method, the completing the square guide goes through it step by step. Try matching a quadratic to its graph, then press Mark to check in the Graphing Functions course.

Quick check

Reading vertex form

• The vertex is (h, k), with the sign of h flipped from the bracket: (x − 3) means h = 3, and (x + 1) means h = −1.
• a is the same as in standard form. It sets the direction (up if positive) and how steep the parabola is.
• The axis of symmetry is x = h.
• The minimum or maximum value of y is k.
• It is the parent graph y = x² moved h units right and k units up, then stretched by a.

Common mistakes

• Getting the sign of h wrong. y = (x + 5)² has its vertex at x = −5.
• Forgetting to multiply by a when you take the squared number back out of the bracket.
• Halving b before taking out a. Take a out first, then halve the coefficient inside the bracket.
• Changing a. The a in front of the bracket is the same as in ax² + bx + c.
• Rounding h before finding k. Keep fractions exact, such as h = 5/4.

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