How to make previous inputs progressively fade out in a Matlab plot when I add new inputs

You can do this by modifying the 4th Color attribute of past lines.

Here's a demo resulting gif, where I faded out 10% of the transparency each frame, so only the most recent 10 lines are visible.

Here is the code, see my comments for details:

% Set up some demo values for plotting around a circle
a = 0:0.1:2*pi; n = numel(a);
[x,y] = pol2cart( a, ones(1,n) );

% Initialise the figure, set up axes etc
f = figure(1); clf; xlim([-1,1]); ylim([-1,1]);
% Array of graphics objects to store the lines. Could use a cell array.
lines = gobjects( 1, n );
% "Buffer" size, number of historic lines to keep, and governs the 
% corresponding fade increments.
nFade = 10;

% Main plotting loop
for ii = 1:n
    % Plot the line
    lines(ii) = line( [0,x(ii)], [0,y(ii)] );
    % Loop over past lines.
    % Note that we only need to go back as far as ii-nFade, earlier lines
    % will already by transparent with this method!
    for ip = max(1,ii-nFade):ii
        % Set the 4th Color attribute value (the alpha) as a percentage
        % from the current index. Could do this various ways.
        lines(ip).Color(4) = max( 0, 1 - (ii-ip)/nFade );
    end
    % Delay for animation
    pause(0.1);
end

You may want to do some plot/memory management if you've got many lines. You can delete transparent lines by adding something like

if lines(ii).Color(4) < 0.01
    delete(lines(ii));
end

Within the loop. This way your figure won't have loads of transparent remnants.

---

Notes:

• I generated the actual gif using imwrite in case that's of interest too.
• Apparently the 4th Color value 'feature' has been depreciated in R2018b (not sure it was ever officially documented).

Got enough upvotes to motivate making a slightly more fun demo...

Solution for Matlab 2018a or later (or earlier, later than 2012a at least)

Since the fourth color parameter as alpha value is no longer supported in Matlab 2018a (and apparently was never supposed to as Cris Luengo pointed out), here a solution that works in Matlab 2018a using the patchline function from the file exchange (credits to Brett Shoelson).

% init the figure
figure(); axes();
hold on; xlim([-1 0.5]); ylim([0 1]);

% set fraction of alpha value to take
alpha_fraction = 0.7;
n_iterations = 200;

% looping variable to prevent deleting and calling already deleted lines
% i.e. to keep track of which lines are already deleted
delete_from = 1;

for i=1:n_iterations
    % your x, y data
    [x, y] = doSomething(i);

    % create line with transparency using patchline
    p(i) = patchline(x,y, 'linewidth', 1, 'edgecolor', 'k');

    % set alpha of line to fraction of previous alpha value
    % only do when first line is already plotted
    if i > 1
        % loop over all the previous created lines up till this iteration
        % when it still exists (delete from that index)
        for j = delete_from:i-1       
            % Update the alpha to be a fraction of the previous alpha value
            p(j).EdgeAlpha = p(j).EdgeAlpha*alpha_fraction;

            % delete barely visible lines
            if p(j).EdgeAlpha < 0.01 && delete_from > j
                delete(p(j));
                % exclude deleted line from loop, so edgealpha is not
                % called again
                delete_from = j;
            end
        end
    end
    % pause and behold your mechanism 
    pause(0.1);
end

I included the deletion of barely visible lines, as suggested by @Wolfie (my own, perhaps less elegant implementation)

And here a demonstration of a quick release mechanism:

I'm adding a 2nd answer to clearly separate two completely different approaches. My 1st answer uses the undocumented (and as of 2018b, depreciated) transparency option for lines.

This answer offers a different approach for line drawing which has no compatibility issues (these two 'features' could be implemented independently):

• Create a fixed n lines and update their position, rather than creating a growing number of lines.
• Recolour the lines, fading to white, rather than changing transparency.

Here is the code, see comments for details:

% "Buffer" size, number of historic lines to keep, and governs the 
% corresponding fade increments.
nFade = 100;

% Set up some demo values for plotting around a circle
dt = 0.05; a = 0:dt:2*pi+(dt*nFade); n = numel(a); b = a.*4;
[x1,y1] = pol2cart( a, ones(1,n) ); [x2,y2] = pol2cart( b, 0.4*ones(1,n) ); 
x = [zeros(1,n); x1; x1+x2]; y = [zeros(1,n); y1; y1+y2]; 

% Initialise the figure, set up axes etc
f = figure(1); clf; xlim([-1.5,1.5]); ylim([-1.5,1.5]);

% Draw all of the lines, initially not showing because NaN vs NaN
lines = arrayfun( @(x)line(NaN,NaN), 1:nFade, 'uni', 0 );
% Set up shorthand for recolouring all the lines
recolour = @(lines) arrayfun( @(x) set( lines{x},'Color',ones(1,3)*(x/nFade) ), 1:nFade );

for ii = 1:n
    % Shift the lines around so newest is at the start
    lines = [ lines(end), lines(1:end-1) ]; 
    % Overwrite x/y data for oldest line to be newest line
    set( lines{1}, 'XData', x(:,ii), 'YData', y(:,ii) );
    % Update all colours
    recolour( lines );           
    % Pause for animation           
    pause(0.01);
end

Result: