update example for the change

Author: cynthiajoan

packages/firebase_ai/firebase_ai/example/lib/main.dart

@@ -13,6 +13,7 @@
 // limitations under the License.
 
 import 'package:firebase_ai/firebase_ai.dart';
+import 'package:firebase_app_check/firebase_app_check.dart';
 import 'package:firebase_auth/firebase_auth.dart';
 import 'package:firebase_core/firebase_core.dart';
 import 'package:flutter/material.dart';
@@ -70,10 +71,10 @@ class _GenerativeAISampleState extends State<GenerativeAISample> {
 
   void _initializeModel(bool useVertexBackend) {
     if (useVertexBackend) {
-      final vertexInstance = FirebaseAI.vertexAI(auth: FirebaseAuth.instance);
+      final vertexInstance = FirebaseAI.vertexAI();
       _currentModel = vertexInstance.generativeModel(model: 'gemini-2.5-flash');
     } else {
-      final googleAI = FirebaseAI.googleAI(auth: FirebaseAuth.instance);
+      final googleAI = FirebaseAI.googleAI();
       _currentModel = googleAI.generativeModel(model: 'gemini-2.5-flash');
     }
   }

packages/firebase_ai/firebase_ai/example/lib/pages/chat_page.dart

@@ -57,12 +57,12 @@ class _ChatPageState extends State<ChatPage> {
           : null,
     );
     if (widget.useVertexBackend) {
-      _model = FirebaseAI.vertexAI(auth: FirebaseAuth.instance).generativeModel(
+      _model = FirebaseAI.vertexAI().generativeModel(
         model: 'gemini-2.5-flash',
         generationConfig: generationConfig,
       );
     } else {
-      _model = FirebaseAI.googleAI(auth: FirebaseAuth.instance).generativeModel(
+      _model = FirebaseAI.googleAI().generativeModel(
         model: 'gemini-2.5-flash',
         generationConfig: generationConfig,
       );

packages/firebase_ai/firebase_ai/example/lib/pages/function_calling_page.dart

@@ -236,8 +236,8 @@ class _FunctionCallingPageState extends State<FunctionCallingPage> {
     );
 
     final aiClient = widget.useVertexBackend
-        ? FirebaseAI.vertexAI(auth: FirebaseAuth.instance)
-        : FirebaseAI.googleAI(auth: FirebaseAuth.instance);
+        ? FirebaseAI.vertexAI()
+        : FirebaseAI.googleAI();
 
     _functionCallModel = aiClient.generativeModel(
       model: 'gemini-2.5-flash',

packages/firebase_ai/firebase_ai/example/lib/pages/grounding_page.dart

@@ -75,8 +75,8 @@ class _GroundingPageState extends State<GroundingPage> {
     }
 
     final aiProvider = widget.useVertexBackend
-        ? FirebaseAI.vertexAI(auth: FirebaseAuth.instance)
-        : FirebaseAI.googleAI(auth: FirebaseAuth.instance);
+        ? FirebaseAI.vertexAI()
+        : FirebaseAI.googleAI();
 
     _model = aiProvider.generativeModel(
       model: 'gemini-2.5-flash',

packages/firebase_ai/firebase_ai/example/lib/pages/image_generation_page.dart

@@ -48,8 +48,8 @@ class _ImageGenerationPageState extends State<ImageGenerationPage> {
 
   void _initializeModel() {
     final aiClient = widget.useVertexBackend
-        ? FirebaseAI.vertexAI(auth: FirebaseAuth.instance)
-        : FirebaseAI.googleAI(auth: FirebaseAuth.instance);
+        ? FirebaseAI.vertexAI()
+        : FirebaseAI.googleAI();
 
     _model = aiClient.generativeModel(
       model: 'gemini-2.5-flash-image',