Programmatically adding a Page item to a Wordpress nav menu

aux-functions.php

<?php

  /**
   * Adds Page to a WordPress navmenu
   * @param [int] $page_id    The ID of the page you want to add
   * @param [str] $page_title Title of menu item
   * @param [int] $menu_id    NavMenu ID
   * @param [int] $parent     (Optional) Menu item Parent ID
   */
  function add_page_to_menu($page_id, $page_title, $menu_id, $parent = 0){
    wp_update_nav_menu_item($menu_id, 0, 
      array(  'menu-item-title' => $page_title,
             'menu-item-object' => 'page',
             'menu-item-object-id' => $page_id,
             'menu-item-type' => 'post_type',
             'menu-item-status' => 'publish',
             'menu-item-parent-id' => $parent));
  }