How does the episode with the mad dog change Scout’s perception (opinion) of her father?

We know from the first few chapters of To Kill a Mockingbird that Scout loves her father, but it is a distant sort of relationship.  She respects his intelligence, she loves reading with him, and she respects his authority.  However, we see more interaction between her and Jem, between her and Dill, and even between her and Calpurnia, than we see interactions between her and Atticus. 

At the start of Chapter 10, Harper Lee, speaking through Scout, begins with, "Atticus was feeble: he was nearly fifty."  She then lists things that she sees as shortcomings with him -- he wears glasses, he doesn't "do" anything, he works in an office; and he doesn't hunt, fish, smoke or play poker, all of which are activities that the children of Maycomb see as admirable.  Scout complains to Miss Maudie that "Atticus can't do anything."  Miss Maudie tries to explain that Atticus can do plenty, but Scout is not impressed.

Then the mad dog appears on the street.  Sheriff Tate and Atticus arrive together.  Scout and Jem expect to see the Sheriff deal with the matter, but they are surprised when the Sheriff hands his rifle to Atticus and convinces Atticus to shoot the sick dog ("Mr. Tate handed the rifle to Atticus; Jem and I nearly fainted.")  The children are surprised and awed to see Atticus shoot the dog with a single shot.

After the shot, Scout describes Jem as "paralyzed" with disbelief at what they just witnessed.  Miss Maudie explains to Scout that Atticus was once known as the best shot in Maycomb.  Scout is noticeably proud of Atticus and tells Jem she wants to tell everyone at school about the incident on Monday.  Jem talks her out of it.  We see here that Scout becomes proud of something that her father can "do"; but we also see Jem growing into manhood and recognizing that individuals have secrets about their lives that they keep privately to themselves.  The chapter ends with Jem shouting, "Atticus is a gentleman, just like me."  This incident is as much about Jem's growth and understanding as it is about Scout.