Final answer:
An absolute value equation with an axis of symmetry at x=3 and no x-intercepts can be given by y = |x-3| + 1, where the positive value of k ensures the graph does not touch the x-axis.
Step-by-step explanation:
To create an absolute value equation with an axis of symmetry at x=3 and that also has no x-intercepts, we can consider the nature of absolute value functions.
Absolute value functions are V-shaped, and their axis of symmetry is a vertical line passing through the vertex of the V.
Having no x-intercepts means that the entire graph of the function must be above the x-axis.
The general form of an absolute value function is y = a|x-h| + k, where (h, k) is the vertex of the parabola. To have an axis of symmetry at x=3, we set h to 3.
To ensure no x-intercepts, k must be positive (so the vertex is above the x-axis).
A simple equation that meets these criteria might look like y = |x-3| + 1, where the vertex is (3, 1), indicating an axis of symmetry at x=3 and no x-intercepts since the entire graph is shifted upwards by 1 unit.