some lines may have no intercepts with the x-x− or with the y-y− axis.
this includes lines such as y = 1/xy=1x.
graph{1/x [-5.23, 5.23, -2.615, 2.616]}
there is no point on the graph where x = 0x=0, since 1/010 is undefined. this means that there cannot be a y-y−intercept for this graph.
though the y-y−value does tend to 00 as xx goes to the far right or far left (to -oo−∞ or oo∞), yy never reaches 00, since there is no number that you can divide 11 by to get 00.
since there is no point on the graph where y = 0y=0, there is no x-x−intercept for this graph.
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graphs where an x-x− or y-y− value is constant will have one intercept.
if the x-x− value is constant, and xx is not 00, then there will only be a x-x− intercept (where y = 0y=0, and xx is the constant).
if the y-y− value is constant, and yy is not 00, then there will only be a y-y− intercept (where x = 0x=0, and yy is the constant).
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all linear graphs, where y = mx + cy=mx+c and m != 0m≠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 + 3y=x+3 has its x-x−intercept at (-3,0)(−3,0) and its y-y−intercept at (0,3)(0,3).
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all parabolas, where xx has 22 real roots, have 22 x-x−intercepts. they may also have a y-y−intercept.
graph{x^2 - 2 [-10, 10, -5, 5]}
the roots of the graph are the points where yy is 00, and the solutions for xx are the x-x−coordinates at these points.
the graph shown is y = x^2 - 2y=x2−2; its roots are (-sqrt2,0)(−√2,0) and (sqrt2,0)(√2,0)
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there are examples of graphs with many more x-x− and y-y− intercepts.
the last example in this answer will be some with infinite xx-intercepts.
the graphs of y = sin xy=sinx, y = cos xy=cosx and y = tan xy=tanx all repeat periodically. this means that they meet the x-x−axis at set intervals, and at an infinite number of points.
graph{sin x [-10, 10, -5, 5]}
the graph of sin xx, for example, has an x-x−intercept at every 180^@180∘ on the x-x−axis.
