Quadratic Functions

Vertex Form

Converting a quadratic between standard form and vertex form by completing the square.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

The Parabola

A Form That Shows the Vertex Directly

Recall from Grade 8 that a rigid transformation can slide a figure left, right, up, or down. Before reading on, think about y = (x − 3)² + 2 as a transformed version of y = x²: which direction do you think the graph shifted, and to where do you think the vertex — originally at (0, 0) for y = x² — moved?

Definition — Vertex Form

A quadratic function written as y = a(x − h)² + k, where (h, k) is the vertex directly. This form shows y = x² has been shifted h units horizontally and k units vertically, and stretched or reflected according to a.

y = (x − 3)² + 2 shifts the basic parabola y = x² right 3 units (notice the subtraction inside the parentheses shifts right, not left — since it takes x = 3 to make the inside equal 0, matching the original vertex condition) and up 2 units, moving the vertex from (0, 0) to (3, 2).

Converting from standard form to vertex form uses completing the square: rewriting x² + bx as part of a perfect square trinomial by adding and immediately subtracting the same value, (b/2)², so the expression's value doesn't actually change.

Worked Example — Completing the Square to Find Vertex Form

Convert y = x² + 6x + 5 to vertex form. Take half of 6 and square it: (6/2)² = 9. Add and subtract 9 to keep the expression equivalent: y = (x² + 6x + 9) − 9 + 5. The first three terms now form a perfect square: y = (x + 3)² − 4. The vertex is (−3, −4).

Worked Example — Reading the Vertex Directly from Vertex Form

Identify the vertex of y = −2(x − 5)² + 7. Comparing to y = a(x − h)² + k: h = 5 and k = 7, so the vertex is (5, 7) — read directly, with no calculation needed. Since a = −2 is negative, the parabola opens downward.

y = a(x - h)^{2} + k

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

Watch the sign carefully inside the parentheses: y = a(x − h)² + k has vertex (h, k), so y = (x + 3)² actually means h = −3 (since x + 3 is the same as x − (−3)), not h = 3.

Common Mistakes

Reading the vertex's x-coordinate with the wrong sign, such as reading h = 3 directly from y = (x + 3)² instead of h = −3.
Rewrite x + 3 as x − (−3) before matching it to the (x − h) pattern — the vertex form always has a subtraction inside the parentheses, so a plus sign means h is actually negative.
Forgetting to subtract back the value added when completing the square, changing the expression's actual value.
Whatever value is added to complete the square inside the parentheses must be subtracted right back outside it, in the same step — this keeps the expression exactly equivalent to the original.

Key Takeaways

Vertex form, y = a(x − h)² + k, shows a parabola's vertex, (h, k), directly.
Converting standard form to vertex form uses completing the square, adding and subtracting the same value to preserve the expression's value.
A plus sign inside the parentheses of vertex form means h is negative — rewrite as a subtraction before reading off the vertex.

Summary

Vertex form connects a quadratic's equation directly to the transformations that produced its graph. The next unit shifts from graphing quadratic functions to solving quadratic equations, starting with factoring.

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