Quadratics

Vertex and Graph

For f(x)=x²+bx+c find the vertex (h,k) with h=-b/(2a) and k=f(h), the axis of symmetry x=h, and the x-intercepts by solving f(x)=0. All derived quantities are computed exactly.

vertex-and-graph The vertex is the minimum (or maximum) of the parabola. The axis of symmetry passes through it. X-intercepts are the roots of the equation f(x)=0.

Example

Find the vertex, axis of symmetry, and opening direction from coefficients.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

f(x)=x2−4x+3f(x)= x^{2} - 4x + 3

Step 2 — Axis

Find the axis: 4 / 2 = 2.

x=42=2x= \frac{4}{2} = \hl{2}

Step 3 — Vertex y-value

Substitute x = 2: 4 - 8 + 3 = -1.

f(2)=4−8+3=-1f( 2 )= 4 - 8 + 3 = \hl{-1}

Step 4 — Vertex

The vertex is (2, -1).

(2,-1)( \hl{2} , \hl{-1} )

Step 5 — Opening direction

Since a = 1, the parabola opens up.

opens up\hl{opens up}