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Quadratic Equation Graph — Definition, Formula & Examples

A quadratic equation graph is the U-shaped curve (parabola) you get when you plot y=ax2+bx+cy = ax^2 + bx + c on a coordinate plane. It opens upward when a>0a > 0 and downward when a<0a < 0, with its turning point called the vertex.

The graph of a quadratic equation y=ax2+bx+cy = ax^2 + bx + c (where a≠0a \neq 0) is a parabola with vertex at (−b2a,  f ⁣(−b2a))\left(-\frac{b}{2a},\; f\!\left(-\frac{b}{2a}\right)\right), axis of symmetry x=−b2ax = -\frac{b}{2a}, and concavity determined by the sign of aa.

Key Formula

xvertex=−b2ax_{\text{vertex}} = -\frac{b}{2a}
Where:
  • aa = Coefficient of $x^2$; determines width and direction of opening
  • bb = Coefficient of $x$; shifts the vertex left or right

How It Works

To graph a quadratic equation, start by finding the vertex using x=−b2ax = -\frac{b}{2a}, then substitute that xx-value back into the equation to get the yy-coordinate. Draw the axis of symmetry as a vertical dashed line through the vertex. Next, choose a few xx-values on one side of the axis, compute their yy-values, and mirror those points to the other side. Finally, connect the points with a smooth curve. If a>0a > 0 the parabola opens upward (vertex is a minimum); if a<0a < 0 it opens downward (vertex is a maximum).

Worked Example

Problem: Graph the quadratic equation y=x2−4x+3y = x^2 - 4x + 3. Find the vertex, axis of symmetry, and x-intercepts.
Find the vertex x-coordinate: Use the vertex formula with a=1a = 1 and b=−4b = -4.
x=−−42(1)=2x = -\frac{-4}{2(1)} = 2
Find the vertex y-coordinate: Substitute x=2x = 2 into the equation.
y=(2)2−4(2)+3=4−8+3=−1y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1
Find the x-intercepts: Factor the quadratic to find where y=0y = 0.
x2−4x+3=(x−1)(x−3)=0  ⟹  x=1,  x=3x^2 - 4x + 3 = (x - 1)(x - 3) = 0 \implies x = 1,\; x = 3
Answer: The parabola has vertex (2,−1)(2, -1), axis of symmetry x=2x = 2, and crosses the x-axis at (1,0)(1, 0) and (3,0)(3, 0). Since a=1>0a = 1 > 0, it opens upward.

Why It Matters

Graphing quadratics appears throughout Algebra 1, Algebra 2, and the SAT/ACT. In physics, projectile motion follows a parabolic path, so reading these graphs lets you find maximum heights and landing distances directly.

Common Mistakes

Mistake: Forgetting the negative sign in x=−b2ax = -\frac{b}{2a} when bb is already negative, leading to a wrong vertex.
Correction: Substitute the value of bb including its sign. For b=−4b = -4: x=−(−4)2a=42ax = -\frac{(-4)}{2a} = \frac{4}{2a}, not −42a-\frac{4}{2a}.