Question 495651
The graph could be shift up or down such that it never touches x.
Consider the following graphs:
y = x^2 +1.
{{{graph(500,500,-10,10,-10,10, x*x+1)}}}
.
y = -x^2 -1.
{{{graph(500,500,-10,10,-10,10, -x*x-1)}}}
.
The "why" part of your question is because the function is define for all 'x', from -infinity to +infinity.
So, by definition, it is defined for x=0.  That is the y-intercept.
.
As I showed above, you can manipulate the values to avoid the x-axis, at least for even polynomials.  But with odd polynomials, including linear equations (the exponent = 1), there will be an x-intercept.
.
Consider:
y = x^3 + 1
{{{graph(500,500,-10,10,-10,10, x^3+1)}}}