How many intercepts can a line have?

2 Answers
Apr 11, 2018

Answer:

see below

Explanation:

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.

Apr 11, 2018

Answer:

It is possible for a line to have an infinite number of intercepts with the #x# or #y#-axis.

Explanation:

It is possible for a line to have an infinite number of intercepts with the #x# or #y#-axis.

The line #x=0# has an infinite number of intercepts with the #y#-axis.

The line #y=0# has an infinite number of intercepts with the #x#-axis.

Any line of the format

#y=mx+b#

where #mne0# has exactly one #y#-intercept and one #x#-intercept. If #b=0#, then both the #x# and #y#-intercepts are at the origin #(0,0)#.