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Quadratic Functions

Three forms, one parabola — vertex, roots, and symmetry.

Interactive graph

Interactive Lab

Drag corner to resize ↘

Drag the sliders in the Desmos graph. Watch how each parameter reshapes the curve in real time.

VisualCoefficients stare at you directly. The y-intercept is hiding in plain sight.

Algebra $y = ax^2 + bx + c$ . Here $c$ is the y-intercept.

y = ax^2 + bx + c

VisualThe vertex $(h,k)$ sits right in the equation. Drag the red point — watch h and k update live.

Algebra $y = a(x-h)^2 + k$ . Vertex at $(h,k)$ . Axis of symmetry: $x=h$ .

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

VisualThe roots are the equation's gifts — they appear as constants.

Algebra $y = a(x-r_1)(x-r_2)$ . Roots at $x=r_1, r_2$ . Axis of symmetry: $x=\tfrac{r_1+r_2}{2}$ .

y = a(x-r_1)(x-r_2)

VisualA crystal ball under the radical — it tells you how many x-intercepts exist before you find them.

Algebra $\Delta = b^2 - 4ac$ . Positive: two roots. Zero: one (repeated). Negative: none (real).

\Delta = b^2 - 4ac

Vertex form

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

Vertex (h,k)

Axis of symmetry

x = \frac{-b}{2a}

From standard form

Quadratic formula

x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

Roots from standard form

Discriminant

\Delta = b^2 - 4ac

Predicts root count

Mark this lesson complete to update your mastery.

easyquadratics

Find the vertex of

y=(x-3)^2+4

easyquadratics

Find the y-intercept of

y=x^2-5x+6

mediumquadratics

Convert

y=x^2+6x+5

to vertex form.

mediumquadratics

Find the roots of

y=x^2-5x+6

hardquadratics

For what value of

k

does

y=x^2+kx+9

have exactly one real root?