1Cademy - Graphing $$y = x^2

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Finding the Axis of Symmetry and Vertex of a Parabola

Example

Graphing $y = x^2 - 6x + 8$

To graph the parabola $y = x^2 - 6x + 8$ :

Step 1: The equation is already written with $y$ on one side. Step 2: The coefficient $a = 1$ is positive, so the parabola opens upward. Step 3: Find the axis of symmetry using $x = -\frac{b}{2a}$ . Since $b = -6$ and $a = 1$ , $x = -\frac{-6}{2(1)} = 3$ . The axis of symmetry is $x = 3$ . Step 4: Find the vertex by substituting $x = 3$ into the equation: $y = (3)^2 - 6(3) + 8 = -1$ . The vertex is $(3, -1)$ . Step 5: Find the $y$ -intercept by setting $x = 0$ : $y = 0^2 - 6(0) + 8 = 8$ . The $y$ -intercept is $(0, 8)$ . The point symmetric to the $y$ -intercept across the axis of symmetry $x = 3$ is $(6, 8)$ . Step 6: Find the $x$ -intercepts by setting $y = 0$ : $0 = x^2 - 6x + 8$ . Factoring gives $0 = (x - 2)(x - 4)$ , so $x = 2$ and $x = 4$ . The $x$ -intercepts are $(2, 0)$ and $(4, 0)$ . Step 7: Graph the parabola by plotting the vertex, intercepts, and the symmetric point, and connecting these five points with a smooth curve.

Updated 2026-04-21

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OpenStax Elementary Algebra

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