How do you identify the vertex for the graph of the quadratic equation y = 2x^2 + 12x + 3?

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Identifying the Vertex of the Quadratic Equation

To find the vertex of the quadratic equation y = 2x2 + 12x + 3, we can use the vertex formula. The vertex of a parabola represented by the equation y = ax2 + bx + c can be found using the coordinates:

  • x-coordinate: x = -b / (2a)
  • y-coordinate: Substitute the x-coordinate back into the equation.

Step 1: Identify a and b

In our equation, a = 2 and b = 12.

Step 2: Calculate x-coordinate

Using the formula for the x-coordinate:

x = -12 / (2 * 2) = -12 / 4 = -3

Step 3: Calculate y-coordinate

Now, we’ll substitute x = -3 back into the original equation to find the y-coordinate:

y = 2(-3)2 + 12(-3) + 3

y = 2(9) - 36 + 3

y = 18 - 36 + 3

y = -15

Final Vertex Coordinates

Thus, the vertex of the parabola is located at: (-3, -15).

Understanding that the vertex is the point where the parabola reaches its maximum or minimum value helps in graphing and analyzing the quadratic function. In this case, since the coefficient of x2 (which is 2) is positive, the vertex represents the minimum point of the parabola.

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