1Cademy - Graphing $$y = 2x^2 - 4x

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Graphing a Quadratic Equation in Two Variables

Example

Graphing $y = 2x^2 - 4x - 3$

To graph the parabola $y = 2x^2 - 4x - 3$ , apply the systematic graphing procedure:

Step 1: The equation is already written with $y$ on one side.

Step 2: The coefficient $a = 2$ is positive, so the parabola opens upward.

Step 3: Find the axis of symmetry using $x = -\frac{b}{2a}$ . Since $b = -4$ and $a = 2$ , $x = -\frac{-4}{2(2)} = -\frac{-4}{4} = 1$ . The axis of symmetry is the line $x = 1$ .

Step 4: Find the vertex by substituting $x = 1$ into the equation: $y = 2(1)^2 - 4(1) - 3 = 2(1) - 4 - 3 = 2 - 4 - 3 = -5$ . The vertex is the point $(1, -5)$ .

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

Step 6: Find the $x$ -intercepts by setting $y = 0$ : $0 = 2x^2 - 4x - 3$ . This quadratic does not factor, so use the Quadratic Formula with $a = 2$ , $b = -4$ , and $c = -3$ :

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-3)}}{2(2)}$

Simplify under the radical:

$x = \frac{4 \pm \sqrt{16 + 24}}{4}$ $x = \frac{4 \pm \sqrt{40}}{4}$

Simplify the radical ( $\sqrt{40} = 2\sqrt{10}$ ):

$x = \frac{4 \pm 2\sqrt{10}}{4}$

Factor out $2$ from the numerator to reduce the fraction:

$x = \frac{2(2 \pm \sqrt{10})}{4} = \frac{2 \pm \sqrt{10}}{2}$

The approximate values of the $x$ -intercepts are (2.5, 0) and $(-0.6, 0)$ .

Step 7: Graph the parabola by plotting the vertex, the $y$ -intercept, the symmetric point, and the $x$ -intercepts, then connecting these points with a smooth, upward-opening curve.

Updated 2026-04-21

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OpenStax Elementary Algebra

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