The function you are referring to is f(x) = x² + 1, which is a quadratic function. When we talk about the axis of symmetry for a quadratic function, we are looking for a vertical line that passes through the vertex of the parabola described by the function.
To identify the axis of symmetry mathematically, we can use the formula:
x = -b / (2a)
In this case, the function can be rewritten in standard form as f(x) = ax² + bx + c, with a = 1, b = 0, and c = 1. Plugging these values into the formula gives us:
x = -0 / (2 * 1) = 0
This means the axis of symmetry for this function is the line x = 0. Graphically, this would appear as a vertical line that intersects the x-axis at the point (0, 0).
Now, if you were to sketch the graph of f(x) = x² + 1, you would see a parabola that opens upwards, with its vertex at the point (0, 1), since the minimum value occurs at x = 0 and f(0) = 1. The axis of symmetry, therefore, divides the parabola into two mirror-image halves.
In summary, for the function f(x) = x² + 1, the graph showing the axis of symmetry is the vertical line represented by x = 0, which runs through the vertex of the parabola and indicates that the graph is symmetric about this line.