Parabolas

Parabola Vertex Form

Rewrite a quadratic in vertex form by completing the square.

parabola-vertex-form Completing the square rewrites y = ax^2 + bx + c as y = a(x-h)^2 + k, where (h, k) is the vertex.

Example

Complete the square to identify and plot the vertex.

highlighted = computed this step

Step 1 — Start equation

Start with the quadratic.

y=x2−4x+1y= x^{2} - 4 x + 1

Step 2 — Complete the square

Complete the square for the x terms.

x2−4x+4=(x−2)2x^{2} -4 x + \hl{4} = (x- \hl{2} )^{2}

Step 3 — Vertex form

Write vertex form.

y=(x−2)2−3y= (x- \hl{2} )^{2} - \hl{3}

Step 4 — Plot vertex form

Plot the parabola and vertex.

Coordinate plot\text{Coordinate plot}
Parabola vertex plotParabola with vertex V(2, -3) opening up.V

Step 5 — Vertex and direction

State the vertex and opening direction.

V=(2,-3)opens upV=( \hl{2} , \hl{-3} ) \quad \text{opens up}