am I correct?
f(x) = x² + 6x + 5
What is the y- intercept of this function? 
My ans is (5,0) 
am I correct?

------------------------------------------------

No.  You have it backwards.  It's (0,5).  Remember, it's x that 
you replace by 0 to find the y-intercept.  You do know, don't you, 
that f(x) is exactly the same thing as y? 

In other words there is no difference between

f(x) = x² + 6x + 5 

and

   y = x² + 6x + 5

f(x) is only a different way of writing y. So when you
plug in 0 for x, you get
 
   y = 0² + 6(0) + 5

   y = 5

So it's (0,5) [and NOT (5,0)! That's important!] 

Remember, that x comes before y in the alphabet, and you
can use that fact to remember that what you substitute
for x comes first and what you get for y, comes second
in the ordered pair that represents the point.  It is
important not to get these backward.

-----------------------------------------

and what is the zero's of this parabola? 
My ans is (0,-1),(0,-5) 

-----------------------------------------

That's wrong, You have those backwards too.
Remember that x comes before y in the alphabet.
It's (x,y), and not (y,x)! So the x-intercepts
are the points (-1,0) and (-5,0).  But that's
not all you are getting wrong. You are calling 
points represented by ordered pair "zeros". 
This is wrong, too. 

Now let's go through this and see if I can
straighten you out.  Your difficulty is not
with the algebra manipulation. You do that 
correctly. Your difficulty is with the words
and what they mean, and with the notation.

The x-intercepts are found by substituting 0 
for f(x) or y, and solving.

You apparently did this correctly:

                   y = x² + 6x + 5
                   0 = x² + 6x + 5

It looks better when you put the = 0 on the right
instead of the left:

             x² + 6x + 5 = 0
 
          (x + 1)(x + 5) = 0
  
Setting x + 1 = 0 gives x = -1
Setting x + 5 = 0 gives x = -5

But you got the coordinates backward.  Notice that 
the value you substituted for y, namely 0,
is supposed to be written SECOND and the x-value is 
written FIRST.  So the x-intercepts are (-1,0) and 
(-5,0). Don't get the x-value and the y-value 
backward.  When you substitute 0 for x, you write 0 
first like this (0,___) and you then fill in the blank 
with what you end up getting for y. And when you 
substitute 0 for y, you write 0 second like this 
(___, 0) and you fill in the blank with what you end 
up getting for x. 

Now also I need to tell you the difference between "zeros" 
and "x-intercepts".  They are certainly related! But a zero
of a function is just a SINGLE NUMBER.  It's not a point 
and it's not an ordered pair like (-1,0) and (-5,0). What 
we call the "zeros" are just the x-coordinates of these 
x-intercept points, not the whole points.  The "zeros" are 
just the numbers -1 and -5, not the x-intercept points
(-1,0) and (-5,0).  You see the difference, right?  Yes,
math is picky!  :-)  

I hope this helps you get these things straight.

Edwin