A circle has the equation $(x-4)^{2}+(y+7)^{2}=10$. Find the center, radius, and intercepts of the circle and then sketch the graph of the circle.

Step 1 ：The equation of a circle is given by \((x-h)^2 + (y-k)^2 = r^2\) where \((h, k)\) is the center of the circle and \(r\) is the radius. Comparing this with the given equation, we can see that the center of the circle is \((4, -7)\) and the radius is \(\sqrt{10}\).

Step 2 ：To find the x-intercepts, we set \(y = 0\) in the equation and solve for \(x\). Similarly, to find the y-intercepts, we set \(x = 0\) in the equation and solve for \(y\).

Step 3 ：The x-intercepts and y-intercepts are complex numbers, which means the circle does not intersect the x-axis or y-axis.

Step 4 ：Final Answer: The center of the circle is at point \((4, -7)\), the radius is \(\sqrt{10}\), and there are no real x-intercepts or y-intercepts. The circle does not intersect the x-axis or y-axis. The graph of the circle would be a circle centered at point \((4, -7)\) with a radius of \(\sqrt{10}\), and it would not intersect the x-axis or y-axis.

Step 5 ：\[\boxed{\begin{align*} \text{Center} & : (4, -7) \\ \text{Radius} & : \sqrt{10} \\ \text{X-intercepts} & : \text{None} \\ \text{Y-intercepts} & : \text{None} \end{align*}}\]