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Vertex

Quadratic functions

Japanese school year: Math I

What you learn

You will learn how to determine the vertex and axis of symmetry of a parabola by completing the square. This is crucial for sketching graphs and finding the maximum or minimum values of quadratic functions. Prior knowledge of algebraic expansion and factoring is recommended for smooth learning.

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Key points

This vertex form immediately reveals the vertex coordinates (p,q)(p, q) of the parabola. It is used to sketch the graph and determine maximum or minimum values.

y=a(x−p)2+q  ⟹  (p,q)y = a(x - p)^2 + q \implies (p, q)

Completing the square into a ()2( )^2 form rewrites the general quadratic expression so you can easily read the vertex and axis of symmetry.

y=ax2+bx+c=a(x+b2a)2−b2−4ac4ay = ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2 - 4ac}{4a}

This formula calculates the vertex coordinates of y=ax2+bx+cy = ax^2 + bx + c directly from coefficients a,ba, b, and cc, skipping the step of completing the square.

(−b2a,  −b2−4ac4a)\left(-\frac{b}{2a}, \; -\frac{b^2 - 4ac}{4a}\right)

This line equation represents the axis of symmetry dividing the parabola symmetrically in half, matching the xx-coordinate of the vertex.

x=−b2ax = -\frac{b}{2a}

Choose a set to practice.