From the Super Mario Wiki
A Frog Suit is an item power-up first debut in Super Mario Bros. 3. The Frog Suit transforms Mario or Luigi into Frog Mario or Frog Luigi. It grants Mario or Luigi the power to jump higher, swim faster, resist water currents better, and hop while walking on the ground. When Frog Mario or Frog Luigi is hit by an enemy, they revert back to Super Mario or Super Luigi.
Super Mario Bros. 3
The Frog Suit first appears in Super Mario Bros. 3. It is obtainable by accessing certain secret areas and hitting a massive question block, as well as Toad Shops from World 3 and beyond. The suit, as stated above, lets Mario swim faster and better, jump higher, swim through water currents, and hop while on land.
The Frog Suit is a handy item to use in underwater levels. Without one, Mario and Luigi will sink underwater and can only swim diagonally up and to the left, diagonally up and to the right, and straight up. Frog Suits allow them to remain stationary in the water, and swim in any direction. However, Frog Suits don't provide much help on dry land. When out of water, Frog Mario and Frog Luigi move very slowly, as they can only move by hopping (instead of running). On the positive side, Frog Suits do allow the Mario Brothers to jump higher than they can normally, and it is possible to run on dry land with a Frog Suit if the player is carrying a Koopa Shell.
If Mario or Luigi wears the Frog Suit and defeats a Koopaling, the king will say, "Oh me, oh my! You've been transformed! Shall I change you back with this wand?"
The Adventures of Super Mario Bros. 3
The Frog Suit also appears in quite a good amount of The Adventures of Super Mario Bros. 3 episodes.
In "Oh, Brother!", Luigi used a Frog Suit to swim through a flooded Pipe Maze in order to rescue Mario from Bowser and Kooky. In "Mush-Rumors", Mario and Luigi donned Frog Suits in an attempt to save Norman's family from the Koopas, but were unsuccessful in the end.
The most prominent appearance of the Frog Suit in the cartoon was in the episode "The Ugly Mermaid," because Mario, Luigi, the Princess and Toad were all using Frog Suits (and thus creating Frog Princess and Frog Toad), and they kept those suits through the whole episode.
In the beginning of "7 Continents for 7 Koopas," Mario and Luigi can be seen crossing a bridge wearing Frog Suits. They were planning to relax in the water with Princess Toadstool and Toad, when King Koopa attacked, thus starting the problems for the Mario Bros.
In "Super Koopa", Bowser had access to a Power Pendant that gave him the same powers that Mario and Luigi normally used, including a Frog Suit. Mario and Luigi later wore Frog Suits, which they used to leap away from Super Koopa's attacks.
Additionally, Luigi is seen wearing a Frog Suit during the show's introductory sequence, in which he rescues Toad from Bully and Big Mouth Koopa after they capture him while underwater. Artwork of Frog Luigi also appears on the cover of Shout! Factory's Complete Series DVD set.
Nintendo Comics System
Nintendo Adventure Books
Early on in the Nintendo Adventure Book Leaping Lizards, Luigi can acquire a Frog Suit, which he needs if he encounters a swarm of Bloopers in a body of water in Iggy Koopa's secret lab; the Frog Suit allows Luigi to deflect some Sleepy Ink they spray at him. In a later book, Unjust Desserts, Mario mentions he should have brought a Frog Suit while tumbling through some of the fluid in Yoshi's body.
Paper Mario: Sticker Star
After an approximate 20-year absence, the Frog Suit returns in Paper Mario: Sticker Star for the Nintendo 3DS, mainly as a sticker that Mario can wear. Mario can then avoid enemies' attacks by jumping over them. It was originally set to act like the Multibounce Badge, but that place was taken by the Line Jump sticker.
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