Why do bond prices go down if interest rates go up?

Because it will be traded at a discount so the coupon rate will be lower than the interest rate

Because there is an inverse relationship between coupon reate  and bond price 

if interest rates were to fall after your purchase, the value of your bond would rise because investors cannot buy a new issue bond with a coupon as high as yours.

Bond prices are generally fixed at a set rate. If interest rates go up then the interest on your bond suddenly becomes less attractive. In order to attract investors bond prices need to come down in order to attract a similar yiled. 

Bonds are investment in debt securities which offer a fixed rate of intrest (well mostly fixed, there are floating coupon bonds as well). If the prevailing interest rate is goes up, I would want to invest in the bonds offering this current rate rather than existing bond offering a lower interest rate. To buy the bond offering lower rate I would want to buy it at an even lower rate. Hence, the price of the bond would go down. 

Similary, when the interest rates go down, bond prices will go up.

when stated rate is lower than market rate then the bond prices go down and investor is offered an incentive to buy bonds but interest payments will be lower than current market interest rate.

bond and price are inverse relationship bond price goes bearish then interest rates are bullish vice versa