1Cademy - Graphing $$y = -3x^2 + 12x

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Graphing a Quadratic Equation in Two Variables

Example

Graphing $y = -3x^2 + 12x - 12$

To graph the parabola $y = -3x^2 + 12x - 12$ , apply the standard graphing procedure:

Step 1: The coefficient $a = -3$ is negative, so the parabola opens downward.

Step 2: Find the axis of symmetry using $x = -\frac{b}{2a}$ . Since $b = 12$ and $a = -3$ , $x = -\frac{12}{2(-3)} = -\frac{12}{-6} = 2$ . The axis of symmetry is the line $x = 2$ .

Step 3: Find the vertex by substituting $x = 2$ into the equation: $y = -3(2)^2 + 12(2) - 12 = -12 + 24 - 12 = 0$ . The vertex is the point $(2, 0)$ .

Step 4: Find the $y$ -intercept by setting $x = 0$ : $y = -3(0)^2 + 12(0) - 12 = -12$ . The $y$ -intercept is $(0, -12)$ . The point symmetric to the $y$ -intercept across the axis of symmetry $x = 2$ is $(4, -12)$ .

Step 5: Find the $x$ -intercepts by setting $y = 0$ : $0 = -3x^2 + 12x - 12$ . Factoring out the greatest common factor gives $0 = -3(x^2 - 4x + 4)$ , which is a perfect square trinomial: $0 = -3(x - 2)^2$ . Solving yields $x = 2$ . The only $x$ -intercept is $(2, 0)$ , which is the same as the vertex.

Step 6: Graph the parabola by plotting the vertex, the $y$ -intercept, and the symmetric point, then connect them with a smooth, downward-opening curve.

Updated 2026-04-21

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OpenStax Elementary Algebra

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