DURATION

I was reading the answer that is included in the new pilot paper answer to question four (the question was phrased as such “Discuss how useful duration is as a measure of the sensitivity of a bond price to changes in interest rates”) which contains this statement:

However, duration is only useful in assessing small changes in interest rates because of convexity. As interest rates increase the price of a bond decreases and vice versa, but this decrease is not proportional for coupon paying bonds, the relationship is non-linear. In fact the relationship between the changes in bond value to changes in interest rates is in the shape of a convex curve to origin, see below.

Please could you explain to me the above paragraph. What’s this convexity thing? I don’t understand. I understand that the duration assumes a linear relationship while in reality it’s non linear, however, aren’t all bond coupon paying bonds? The examiner mentions “coupon paying bonds”. What’s this? Please explain the above paragraph.

Also, this line ” This is because it is an average measure based on the gross redemption yield (yield to maturity)” is elusive. Please explain