Edit in JSFiddle

var inputForm;
$(function() {
  var InputForm = {
    $area: null,
    init: function(selector) {
      this.$area = $(selector);
      this.bindEvents();
      return this;
    },
    bindEvents: function() {
      $(document).on('click', '.glyphicon-chevron-down', function(e) {
        this.showInputArea(e);
      }.bind(this));
      $(document).on('click', '.glyphicon-chevron-up', function(e) {
        this.hideInputArea(e);
      }.bind(this));
    },
    showInputArea: function(e) {
      this.getInputArea(e).slideDown();
      $(e.target).addClass('glyphicon-chevron-up').removeClass('glyphicon-chevron-down');
    },
    hideInputArea: function(e) {
      this.getInputArea(e).slideUp();
      $(e.target).addClass('glyphicon-chevron-down').removeClass('glyphicon-chevron-up');
    },
    getInputArea: function(e) {
      return $(e.target).parent().parent().next();
    }
  };

  inputForm = InputForm.init("form");
});

<br>
<form>
  <div class="col-xs-9">
    <p class="e_form_title">
      <label>説明文
        <span class="glyphicon glyphicon-chevron-down" aria-hidden="true"></span>
      </label>
    </p>
    <p style="display: none;">
      <textarea rows="10" cols="50" name="registration_information"></textarea>
    </p>
    <input type="submit" value="登録">
  </div>
</form>