First, let's find the vertex. For a quadratic function ax² + bx + c, the x-coordinate of the vertex is negative b divided by 2a. In our function, a equals 2 and b equals negative 8, so x vertex equals negative negative 8 divided by 2 times 2, which equals 8 divided by 4, equals 2. Substituting x equals 2 into the function gives y equals 2 times 4 minus 16 plus 6, which equals negative 2. Therefore, the vertex is at point (2, -2).
The axis of symmetry is the vertical line that passes through the vertex. Since the vertex has an x-coordinate of 2, the equation of the axis of symmetry is x equals 2. This line divides the parabola into two symmetric halves.
To find the x-intercepts, we set g(x) equal to zero. Solving the equation 2x² - 8x + 6 = 0. First, divide by 2 to get x² - 4x + 3 = 0. Factoring gives us (x - 1)(x - 3) = 0, so x = 1 or x = 3. Therefore, the x-intercepts are (1, 0) and (3, 0).
Finally, let's find the y-intercept. Setting x equal to 0, we get g(0) equals 6, so the y-intercept is (0, 6). To summarize: this quadratic function has vertex at (2, -2), axis of symmetry x = 2, x-intercepts at (1, 0) and (3, 0), and y-intercept at (0, 6). These characteristics completely describe the shape and position of the parabola.