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Medium

Graphing y = x² + bx + c

Simple Explanation

When a = 1, the graph of y = x² + bx + c is an upward-opening parabola (U-shaped) with a minimum point (the vertex). Its axis of symmetry is a vertical line through the vertex, and the graph is a mirror image of itself across that line.

Why Do We Need It?

This is the simplest quadratic graph shape — recognizing it as always U-shaped and upward when a = 1 gives you an instant mental picture before plotting a single point.

See It

Graph of y = x² − 2x − 3
-6-6-4-4-2-22244660xy(1, -4)

An upward-opening parabola with vertex at (1, -4), crossing the x-axis at x = -1 and x = 3

Formula

Vertex Formula of a Quadratic Function

Vertex = (−b/2a, f(−b/2a)), for y = ax² + bx + c

Gives the exact coordinates of the turning point (vertex) of a parabola directly from its coefficients, without needing to complete the square.

a, b, c
— the coefficients of y = ax² + bx + c (a ≠ 0)
−b/2a
— the x-coordinate of the vertex, and the equation of the axis of symmetry
f(−b/2a)
— the y-coordinate of the vertex — the minimum or maximum value of the function

When to use it: Whenever you need the highest or lowest point of a parabola, or its axis of symmetry, directly from the equation y = ax² + bx + c.

Worked Example

Find the vertex of an upward parabola

Find the vertex of y = x² − 2x − 3.

    Why Does This Work?

    Since a = 1 is positive, the squared term always adds a nonnegative amount as x moves away from the vertex in either direction, so the graph always curves upward from its lowest point — never downward.

    Real-Life Example

    The shape of a suspension cable

    A cable hanging under a uniformly distributed load (like a suspension bridge) approximately forms an upward-opening parabola.

    The lowest point of the cable is exactly the vertex of a parabola shaped like y = x² + bx + c — engineers use this vertex to find the cable's lowest sag point.

    Practice

    What is the vertex of y = x² − 4x + 1?

    Medium

    Common mistake

    Forgetting the negative sign inside the vertex formula, x = −b/2a — for y = x² − 2x − 3 (b = −2), the x-coordinate is −(−2)/2 = 1, not −2/2 = −1.

    Quick Review

    • When a = 1, y = x² + bx + c always opens upward (U-shaped), with a minimum vertex.
    • Vertex x-coordinate: x = −b/2a. Substitute back in to find the y-coordinate.
    • The graph is symmetric about the vertical line through the vertex.
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