Disclaimer: I don't own Twilight.
The first time Jasper Hale lays eyes on Bella Swan is in the lunchroom. He barely takes notice of her, until Edward suddenly announces he can't read her mind.
Jasper is put off by that, but finds it amusing that his brother in all but blood has finally found a mind he can't read. It isn't until later that day, after Edward had already fled to Denali, that he learns Bella is Edward's singer. Jasper won't say it, but he silently resents Bella Swan for driving his brother away.
The second time Jasper sees Bella is when his brother jumps to save her from being crushed by Tyler Crowley's van. Jasper is appalled that Edward would jeopardize their family over a human, and, although, he hates to do it, he is already making plans to kill Bella. It is only at Alice's assuring words that everything will be ok that he relents. Bella doesn't know how close she came to death.
The third time, Jasper sees Bella on the school field trip. He finds her pleasant enough, but he's still wary of her.
The fourth time Jasper sees Bella is when Edward brings her to their home. At first Jasper is fully against bringing the human into their home full of vampires, viewing it as an unnecessary risk, but Alice once again assures him that it will be fine. Jasper trusted his wife's judgment, so he goes along with it, but he still keeps his distance.
When James and his coven interrupt his family's baseball game, he knows theirs going to be trouble. Bella's scent is so sweet that it's enough to send any vampire into a frenzy. Jasper is unfortunately right when it's revealed James plans to hunt Bella. Jasper is a former soldier, and is already coming up with a plan of action.
Alice and Jasper go on the run with Bella, and for the first day it all goes well, but then on the second day, as they arrive at the Phoenix airport, Bella somehow manages to escape from him and Alice. Jasper won't ever admit it, but he's impressed.
When they find Bella broken on the floor of the ballet studio, his first thought was if James bit her. When he realizes that he had, in fact, bit her; Jasper thanks his lucky stars that they are in Phoenix, Arizona instead of Forks, Washinton. Bella's disappearance will be so much easier to explain than if it were at home, but then his brother does something he never in a million years expected; he sucked the venom out. Not even Jasper would have thought of that one.
Jasper isn't aware of when he stops seeing Bella has just a human, but instead a member of his family; which, is why he feels so guilty when he snaps at Bella on her birthday. Jasper expects Bella to hate him, but instead, all he feels is understanding and love. Bella doesn't know it, but it was at that moment that Jasper Hale starting loving her as a sister.
When Jasper and his family return to Forks, he is surprised to learn that his new little sister had become friends with werewolves. He can only pray that Bella will come to her senses and drop the wolves as friends.
When Jasper hears about the unexplained deaths in Seattle, he once again knows there is going to be trouble; this scenario is all too familiar. It isn't long before his worst fears are confirmed. Victoria has created a newborn army to destroy Bella and their family.
What Victoria doesn't know, is that he is trained in dealing with newborns, and they have the wolves. It's by luck that they win, and survive the Volturi,. It's only later that he learns what Bella did, and again, he feels his respect for her grow.
On Edward and Bella's wedding day, he feels a surge of pride for his brother and happiness at the fact that he now has a sister. However, only three days later, that happiness ends when he hears the news. Bella's pregnant.
Jasper stays quiet as his wife and brother argue with Bella to terminate the pregnancy. Jasper, however, is conflicted when it comes to his own opinion. On one hand, he understands where his wife and brother are coming from; on the other hand, he respects Bella's decision and finds it honorable.
In Jasper's original time, women died from childbirth all the time. It was just a fact of life, his own mother died giving birth to him. Jasper knows that if Bella dies Edward dies, but Bella is determined to have that baby and survive, and he believes that if anyone can survive this, Bella can.
It is three days later when it happens. Jasper is in his study when he hears Bella's spine snap and the yells and shouts; but all he focuses on Bella's desperate heart trying to keep up. He keeps count of how many thumps he hears, until he can no longer hear her heart.
A deep sorrow feels his soul, a pain he can't describe when the realization that Bella was dead registered. Jasper feels an emptiness in his heart, but he doesn't have time to mourn when the wolves come to kill Renesmee.
Jasper feels nothing but anger. How dare those animals try to destroy the baby that his sister gave her life for, how dare they spit on her sacrifice like it's nothing? Jasper, for once, wants nothing more than to destroy those stupid wolves and put them in their place. As the fight rages on and just when he thinks they're at a stalemate; Jacob reveals that he's imprinted.
Jaspers first thought is on who, and then he understands. Jacob imprinted on Renesmee.
Three days later Bella awakens, and, to Jasper's surprise, she is the calmest newborn he's ever seen.
When Bella comes back from her first hunt, Jasper turns to her and says, "Welcome home, little sister,"
Bella turns to him and says, "Thank you, big brother,"