Approximate the value of 14−−√ to the nearest tenths place. plot the approximation on the number line.

The approximation for √14 to the nearest tenth is 3.7

There are different methods to approximate a square root: the easiest is obviously the calculator, the second option is a tabular interpolation, otherwise, there is the algorithm.

Let's suppose you don't have anything but a piece of paper and a pen/pencil and you need to apply the algorithm. The steps to follow are the following:

A) Write the number you need to calculate, √14, and divide its digits into pairs starting from the units: in our case 14 is a pair.

B) Start from the first pair from the left (14) and find the greatest integer whose square is less than or equal to your pair:
3² < 14 < 4²
Write the number you found (3) in the suitable space.

C) Square the number you found in step B, write it below the pair you considered and perform a difference: 14 - 9 = 5

D) lower the second pair of the original number and write it next to the remainder found in step C. Since our number was only two-digit long, we lower 00 as if it were 14.00, and we get 500.

E) now there is the most delicate step: double the number you wrote in step B  and write it below the 3 · 2 = 6
Making various attempts, you need to find the greatest digit X so that the result of the multiplication 6X · X is less than or equal to the partial remainder we found in step D.
Since 14 is closer to 16 than to 9, let's start with X = 5:
65 · 5 = 325
66 · 6 = 396
67 · 7 = 469
68 · 8 = 544 > 500!!

F) Write the digit you found in the solution space, write the result of the multiplication under the partial remainder and perform the difference: 500 - 69 = 31

G) repeat step D, E, and F until you finish the pairs or until the desired approximation. In your case, you can stop here: the approximation wanted is 3.7

In the picture attached is shown the complete work. 

For this case we have the following expression:
 √14
 The result in this case is:
 3.741657387
 Rounding to the nearest tenths place:
 3.74
 
 3.74
 See attached image To see the graph in a line.