some lines may have no intercepts with the x- or with the y- axis.
this includes lines such as y = 1/x.
graph{1/x [-5.23, 5.23, -2.615, 2.616]}
there is no point on the graph where x = 0, since 1/0 is undefined. this means that there cannot be a y-intercept for this graph.
though the y-value does tend to 0 as x goes to the far right or far left (to -oo or oo), y never reaches 0, since there is no number that you can divide 1 by to get 0.
since there is no point on the graph where y = 0, there is no x-intercept for this graph.
-
graphs where an x- or y- value is constant will have one intercept.
if the x- value is constant, and x is not 0, then there will only be a x- intercept (where y = 0, and x is the constant).
if the y- value is constant, and y is not 0, then there will only be a y- intercept (where x = 0, and y is the constant).
-
all linear graphs, where y = mx + c and m != 0, either have one intercept with each axis or have one intercept with the origin where both axes cross.
graph{x + 3 [-10, 10, -5, 5]}
the graph y = x + 3 has its x-intercept at (-3,0) and its y-intercept at (0,3).
-
all parabolas, where x has 2 real roots, have 2 x-intercepts. they may also have a y-intercept.
graph{x^2 - 2 [-10, 10, -5, 5]}
the roots of the graph are the points where y is 0, and the solutions for x are the x-coordinates at these points.
the graph shown is y = x^2 - 2; its roots are (-sqrt2,0) and (sqrt2,0)
-
there are examples of graphs with many more x- and y- intercepts.
the last example in this answer will be some with infinite x-intercepts.
the graphs of y = sin x, y = cos x and y = tan x all repeat periodically. this means that they meet the x-axis at set intervals, and at an infinite number of points.
graph{sin x [-10, 10, -5, 5]}
the graph of sin x, for example, has an x-intercept at every 180^@ on the x-axis.