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

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Graphing a Quadratic Equation in Two Variables

Example

Graphing $y = -3x^2 - 6x + 5$

To graph the parabola $y = -3x^2 - 6x + 5$ , apply the systematic 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 $a = -3$ and $b = -6$ , $x = -\frac{-6}{2(-3)} = -\frac{-6}{-6} = -1$ . The axis of symmetry is the line $x = -1$ .

Step 3: Find the vertex by substituting $x = -1$ into the equation: $y = -3(-1)^2 - 6(-1) + 5 = -3(1) + 6 + 5 = -3 + 6 + 5 = 8$ . The vertex is the point $(-1, 8)$ .

Step 4: Find the $y$ -intercept by setting $x = 0$ : $y = -3(0)^2 - 6(0) + 5 = 5$ . The $y$ -intercept is (0, 5). The point symmetric to the $y$ -intercept across the axis of symmetry $x = -1$ is $(-2, 5)$ .

Step 5: Find the $x$ -intercepts by setting $y = 0$ : $0 = -3x^2 - 6x + 5$ . Use the Quadratic Formula with $a = -3$ , $b = -6$ , and $c = 5$ :

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(-3)(5)}}{2(-3)}$ $x = \frac{6 \pm \sqrt{36 + 60}}{-6}$ $x = \frac{6 \pm \sqrt{96}}{-6}$ $x = \frac{6 \pm 4\sqrt{6}}{-6}$

Factor out $2$ from the numerator and denominator to simplify:

$x = \frac{2(3 \pm 2\sqrt{6})}{-6} = \frac{3 \pm 2\sqrt{6}}{-3}$

The approximate values of the $x$ -intercepts are $(-2.6, 0)$ and (0.6, 0).

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

Updated 2026-04-21

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OpenStax Elementary Algebra

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