Is there a good way to join these two objects up? Blender Newb here. So take with a grain of salt. : )

I'm learning blender for my job so I'm trying to 3d out some of our past work. I've got this monument sign I'm trying to do to teach myself.

In the picture I've got you can see there is the horizontal curved blue object and the vertically curved purple. I'm trying to get these two curved object to meet at the yellow line to form a doubly curved edge.

Is there anyway to do this without haven't to piece the vertices together one by one?

Let me know if you need more info.
There are a few approaches I might use.

Boolean modifier is the most straight forward route - make the blue and purple parts larger so they overlap, and then use the intersection operation. A route like this usually works well in CAD software, but it's often not the best in Blender.

A better approach in Blender is using its mesh editing tools like proportional editing (the o key in edit mode).


Start with a plane, rotate it to face the front and press TAB to enter edit mode. 
Use S + X to sale in one axis and S + Y to scale in the other, and G to position it into the right space.
Subdivide it using loop cut, to get a bunch of edges along where it needs to curved (use the scroll where to increase the number of cuts):

Select the one or two vertices in the top middle (using RMB and SHIFT+RMB if selecting two). Press o to use proportional editing mode. Press g to start moving the vertices. Press z to only move on the Z axis. Use the scroll wheel to increase the size of the white circle so you start moving all the vertices at once. You can try SHIFT+o to cycle different fall offs ("Inverse Square" worked best for me):

The bottom is also curved. We'll fix that later.
Now do the same for the other axis from the top view (press 7 on the numpad), but this time select both the top vertices and bottom vertices. Again, use g to grab, and now y to constrain to the Y axis.  
Let's fix the bottom edge. Turn proportional editing off using o. Switch back to front view using 1 on numpad. Select the whole bottom row by holding ALT and clicking and edge (or B and drawing a box around it). Press S to start scaling, z to constrain to the Z axis, and 0 to scale to zero.
The whole thing may need its position fixed now. a to select all and use the on screen widget to move it around one axis at a time


To finish it, and add the ridge at the top, you can use extrusions:


Select the top row using ALT+LMB
Press e followed by y and drag the mouse to extrude outward along the Y axis
Press e followed by z and drag to extrude upward along the Z axis
Press e + y again to extrude it inward