How can I verify if Apple is slowing down my phone because of an old battery?
iOS 11.3 brings iPhone Battery Health (Beta). This new feature made available in this update provides users with info about the health of their battery, allowing them to determine if the condition of their battery is affecting their iPhone's performance.


  
  Displays information on iPhone maximum battery capacity and peak performance capability
  Indicates if the performance management feature that dynamically manages maximum performance to prevent unexpected shutdowns is on and includes the option to disable it
  Recommends if a battery needs to be replaced
  


Go to Settings → Battery → Battery Health (Beta). If your battery is functioning correctly, you will see the left message. If not, you will see the right message and you can tap Disable to disable the performance management.


  
  


You can read more about this feature in the Apple KB article


iPhone Battery and Performance




Without updating to iOS 11.3, you can still determine battery health.

If you don't have one of these devices with at least this version of iOS, your device is not being slowed regardless of the state of the battery, unless you've enabled Low Power Mode either manually or when the battery reaches 20% (and disables at 80%).


iPhone 6, iPhone 6s, iPhone SE with iOS 10.2.1
iPhone 7 with iOS 11.2


If you have one of these devices, you can check if your battery is poor using an app:


 Battery Life


To know more accurately, you can run the Geekbench benchmark.


 Geekbench


Make sure Low Power Mode is disabled, then run the Geekbench benchmark (not the battery benchmark, that's for analysing your battery itself and not its effect on iOS) and note your score. If it's more than 500 points less than the expected number (given below), then this indicates a potential problem.


iPhone 6s: 2500
iPhone 7: 3500