Finding the Vertex and Axis of Symmetry
To find the vertex and axis of symmetry of the given quadratic function, we can follow a systematic approach.
Step 1: Identify the coefficients
In the quadratic function f(x) = 3x² + 12x + 6, we can identify the coefficients:
- a = 3
- b = 12
- c = 6
Step 2: Calculate the x-coordinate of the vertex
The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
Substituting the values of b and a:
x = -12 / (2 * 3)
x = -12 / 6
x = -2
Step 3: Calculate the y-coordinate of the vertex
Now that we have the x-coordinate, we can find the y-coordinate by substituting x = -2 back into the function:
f(-2) = 3(-2)² + 12(-2) + 6
f(-2) = 3(4) – 24 + 6
f(-2) = 12 – 24 + 6
f(-2) = -6
Step 4: State the vertex
Thus, the vertex of the parabola is located at:
(-2, -6)
Step 5: Determine the axis of symmetry
The axis of symmetry for a parabola is a vertical line that passes through the vertex. This line can be expressed as:
x = -2
Conclusion
In summary, for the quadratic function f(x) = 3x² + 12x + 6, the vertex is at (-2, -6) and the axis of symmetry is the line x = -2.