To find the vertex and the axis of symmetry of the quadratic function y = 2x^2 + 2x + 4, we can use the vertex formula.
The vertex of a parabola in the form y = ax^2 + bx + c can be found at the point (h, k), where:
- h = -b / (2a)
- k = f(h)
In our function, a = 2, b = 2, and c = 4.
1. **Calculate h:**
h = -b / (2a) = -2 / (2 * 2) = -2 / 4 = -0.5
2. **Calculate k:**
We need to substitute h back into the equation to find k:
k = 2(-0.5)^2 + 2(-0.5) + 4 = 2(0.25) – 1 + 4 = 0.5 – 1 + 4 = 3.5
So, the vertex is (-0.5, 3.5).
Now, let’s determine the axis of symmetry. The axis of symmetry for a parabola is a vertical line that passes through the vertex and is given by the equation:
x = h
Thus, the axis of symmetry for our function is:
x = -0.5.
In summary, the vertex of the graph of the function y = 2x^2 + 2x + 4 is (-0.5, 3.5), and the axis of symmetry is the line x = -0.5.