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Below is the script and I've also put an image of what it's meant to look like and what my image ends up looking like

// The Landsat 8 top of atmosphere reflectance data
var l8 = ee.ImageCollection("LANDSAT/LC08/C01/T1_TOA");

// ROI

// filter the data collection 
var filtered = l8.filterDate('2013-04-01', Date.now())
        .filterBounds(roi);
print(filtered);

// Get a cloud free image
var cloudFreeImage = filtered.sort('CLOUD_COVER').first();

// add the cloud free true colour image to mapping layer
var rgb_vis = {min: 0, max: 0.3, bands:['B4', 'B3', 'B2']};
Map.addLayer(cloudFreeImage, rgb_vis, 'True color');


// add the cloud-free false-colour image to the mapping layer
Map.addLayer(cloudFreeImage, {min: 0, max: 0.3, bands:['B5', 'B4', 'B3']}, 'False Color');

// add the median image to the mapping layer
Map.addLayer(filtered.median(), rgb_vis, 'median RGB');

// compute NDVI and add the NDVI image to the mapping area
var ndvi = cloudFreeImage.normalizedDifference(['B5', 'B4']);
Map.addLayer(ndvi, {min: 0, max: 1, palette:['brown','yellow','green']}, 'NDVI');

// add a new ndvi band a single l8 image
function addNDVIband(anImage) {
  var ndviTemp = anImage.normalizedDifference(['B5', 'B4']);
  return anImage.addBands(ndviTemp.rename('NDVI'));}

// test the function on a single image
var ndviAdded = addNDVIband(cloudFreeImage);
print ('Before adding NDVI', cloudFreeImage);
print ('After adding NDVI', ndviAdded);

// map the function to the image collection
var ndviMapped = filtered.map(addNDVIband);
print ('After mapping NDVI', ndviMapped);

// apply temporal reducer to the NDVI band that has been added to the image collection. 
Map.addLayer(ndviMapped.median(), {bands: 'NDVI', min: 0, max: 1, palette:['brown','yellow','green']}, 'median ndvi');

/*// chart the ndvi over time
var ndviChart = ui.Chart.image.series({
    imageCollection:ndviMapped.select("NDVI"),
  region: roi,
  reducer: ee.Reducer.median(),
  scale: 90,
  xProperty:'system:time_start'});
print(ndviChart);
*/

[NDVI over region][1

what mine looks like with the code I'm using

2

You haven't said exactly what you want to change about your output, but the two most obvious differences are:

  • The resulting image should be clipped to the polygon. So use Image.clip:

    Map.addLayer(
        ndviMapped.median().clip(roi),
        {bands: 'NDVI', min: 0.1, max: 0.4, palette: ['brown','yellow','green']}, 
        'median ndvi');
    
  • There is a black border. So draw the border using FeatureCollection.style:

    Map.addLayer(ee.FeatureCollection([roi]).style({fillColor: '00000000'}), {}, 'ROI');
    

You may also need to adjust your min and max to get more contrast. You didn't provide your roi polygon so I can't match the coastline, but here's what I get with a simple drawn polygon and min: 0.1, max: 0.4:

Screenshot of modified script

https://code.earthengine.google.com/874abc5de07750eafc07e9bfa4cd13d0

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