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.
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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).
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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)#.
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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)#
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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.
