Am I implementing the convolutional integral correctly?

Hello all,

Actually, I am trying to calculate the convolutional integral of two functions using the following code:

close all;
clear all;
clc;
%set time vector
t = linspace(0,85,16384);
T0=60;
alpha=1;
beta=0.1;

%construct the first part of the convolution
for i=1:size(t,2)
  if(t(i)<=T0)   
    y(i)=alpha/1.5;
  else 
    y(i)=(alpha+beta*(t(i)-T0))/1.5;
  end
end

%plot the first function of the convolution
figure;
plot(t,y,'linewidth',2);
ylim([-1 3.5]);
set(gca,'fontsize',14,'fontweight','bold');
grid on;
xlabel('t');
ylabel('y(t)');
title('First function of the convolution');



%second part of the convolution 
for i=1:size(t,2)
f2(i) = 0.08787*exp(-((t(i)-63.08)/1.593).^2);
end



%plot the second function of the convolution 
figure;
plot(t,f2,'linewidth',2);
legend('f2')
xlim([50 85]);
set(gca,'fontsize',14,'fontweight','bold');
grid on;
xlabel('t');
ylabel('f2(t)');
title('Second function of the convolution');


%compute the convolution using matlab function
z2 = median(diff(t))*conv(y,f2,'same');


%plot the matlab convolution
figure;
plot(t,z2,'linewidth',3);
legend('f2*y')
grid on;
xlabel('t');
ylabel('y_*f');
title('Result of the convolution');
hold on;

The first and second part of my convolutional integral are as follows:

and the result of the convolution is as follows:

But the strange point is that both of my functions (y and f2) are constant until t=60 but the result of the convolution starts from t=20. I don't know what's wrong with my implementation. Does anyone have any idea regarding my code? Am I implementing the convolutional integral correctly?