CUPERTINO, Calif.—Apple Inc. introduced a trio of new iPhones Tuesday, making a bet that some customers will be willing to part with upward of $1,000 for the premium iPhone X while others will also pay up to upgrade regular models.

Inside a theater at its new $5 billion headquarters, Apple worked to conjure the magic that 10 years ago made its iPhone the world’s most valuable company’s most valuable product.

The pricing of the new flagship phone puts it closer to a major appliance or many PCs and represents a test of the enduring cachet of Apple products, already among the priciest in their field.

The products are Apple’s most anticipated in years. The company and its shareholders have endured a slump in iPhone sales growth and the company is facing intensifying competition from Samsung Electronics Co. and others. That anticipation has pushed Apple’s share price up 39% this year to record highs, giving the company an unprecedented market value of more than $830 billion.

The iPhone X boasts advanced features including facial-recognition capability and a 5.8-inch edge-to-edge, high-resolution screen. Apple also unveiled the iPhone 8 and 8 Plus, featuring upgrades to last year’s model with new design and features.

iPhones contributed two thirds of Apple’s $215.64 billion in revenue last fiscal year.

Apple is trying to thread a needle, looking to generate buzz and sales with the iPhone X, which will cost from $999 to $1,149, while still getting people to buy the upgraded phones, which will start at $699 for the iPhone 8 and $799 for the iPhone 8 Plus.

Unveiling three phones—the first time Apple has done so—brings risks with forecasting demand and managing production. In an early sign of the challenges, Apple said the iPhone X won’t start shipping until Nov. 3--much later than usual. The two iPhone 8 models will be available for purchase on Sept. 22, in line with its regular timing for new iPhones.

Apple didn’t give a reason for the delay. The Wall Street Journal reported this month that production glitches in summer delayed the new phone’s production schedule by a month and could lead to extended supply shortfalls.

The iPhone X—which also boasts depth sensors and a dual-lens camera system—arrives amid questions about Apple’s innovative strength. Rivals such as Samsung have matched or leapfrogged ahead of Apple’s smartphone features, introducing larger displays using organic light-emitting diode (OLED) technology and water resistance before Apple did. Meanwhile, rival smartphone makers in China are closing the gap offering similar features at far lower prices.

That has contributed to slower sales growth: Apple’s share of the global smartphone market slipped to 14.5% last year from 19.4% in 2012, according to Strategy Analytics, a market-research firm.

Apple touted the iPhone X, which it pronounces “iPhone Ten,” as the future of the smartphone. “It is the biggest leap forward since the first iPhone,” said Apple Chief Executive Tim Cook.

Some observers were skeptical. Holger Mueller, principal analyst at Constellation Research, said Apple failed to capitalize on the 10-year iPhone anniversary with a revolutionary new smartphone. “Almost all innovations have been shown by the competition, notably Samsung,” he said.

Tuesday’s event was choreographed to connect Apple more deeply with the mystique of its late co-founder Steve Jobs. It took place in a theater bearing his name at a new campus built based on his design concepts. It kicked off with a song by the Beatles, one of his favorite bands, and with audio of remarks he made about Apple’s product philosophy.

“Steve’s vision and passion live on here at Apple Park and everywhere at Apple,” Mr. Cook said.

Apple shares, which had been rising, slid slightly after the company announced the date of the iPhone X’s availability, ending the day down 0.4%.

Other companies have offered facial recognition, but Apple claimed its Face ID system is superior, blending artificial-intelligence with sensors that map faces in 3D using 30,000 invisible infrared dots.

Apple marketing chief Phil Schiller said Face ID can recognize faces in light and darkness, even if the user changes hairstyle or wears glasses or a hat. He said engineers worked to make sure photographs or masks can’t trick it, and said the chances of someone other than the user unlocking the phone with Face ID are one in a million. That compares with one in 50,000 for Touch ID, the current fingerprint sensor technology that sits on a home button Apple is eliminating for the iPhone X.

The iPhone X design has raised hopes that it can reverse Apple’s fortunes in China, Apple’s most important market outside the U.S., where sales have fallen six straight quarters.

“The high-end Chinese phone market is super competitive and customers are very discerning but also enthusiastic,” said Benedict Evans, a partner at Andreessen Horowitz, a venture-capital firm. “If Apple can get something that rings the bell [with them], then this will work.”

The iPhone 8 and 8 Plus feature what Apple has billed as a new design, made of glass on front and back, as well as louder speakers, faster processors and upgraded cameras. All three devices offer wireless charging and new gyroscopes and accelerometers to improve augmented reality experiences.

Apple on Tuesday also announced a new Apple Watch with cellular connectivity, and an Apple TV with ultra-high definition functionality.

The Apple Watch Series 3 will have an LTE chip that enables it to connect to the fastest cellular networks for many of the same functions the iPhone offers. The cellular capabilities allow people to leave their iPhones behind and still send and receive emails and messages by using Siri’s transcription abilities. Users with AirPods wireless headphones could also make and receive calls using the watch.

Apple has sold an estimated 30 million Apple Watches since introducing the device in 2015, according to market research by IDC.

The Apple Watch Series 3 has the potential to extend recent sales gains by boosting the device’s functionality, analysts said.

The new Watch, which packs new chips and new antenna into almost the same size as its predecessor, also will cost more, at $399 with cellular capability or $329 without. The Apple Watch Series 1 will be reduced in price to $249.

The update to Apple’s streaming-media player, called Apple TV 4K, is the first since 2015 and comes as the device has lost market share to lower-priced rivals. Apple Vice President Eddy Cue said Apple will sell ultra-high definition films from Hollywood studios at $19.99, the same price it currently offers high-definition films. There also will be live sports in 4K from ESPN and others.

Write to Tripp Mickle at [email protected]

Corrections & Amplifications Apple Inc. lowered the price on its Apple Watch Series 1 to $249. An earlier version of this article incorrectly said that Apple lowered the price of the Apple Watch Series 2. (Sept. 13, 2017)

QUESTIONS:

Why some experts may believe that Apple failed to capitalize properly on iphone's 10th anniversary?

What are the challenges associated with Apple's decision to unveil three phones?

How would you have your team develop and unveil phones differently in 2017 if you were the CEO of Apple?