How many intercepts can a line have?

Apr 11, 2018

see below

Explanation:

some lines may have no intercepts with the $x -$ or with the $y -$ axis.

this includes lines such as $y = \frac{1}{x}$.
graph{1/x [-5.23, 5.23, -2.615, 2.616]}

there is no point on the graph where $x = 0$, since $\frac{1}{0}$ 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 $- \infty$ or $\infty$), $y$ 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 $y = m x + c$ and $m \ne 0$, 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 $\left(- 3 , 0\right)$ and its $y -$intercept at $\left(0 , 3\right)$.

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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 $\left(- \sqrt{2} , 0\right)$ and $\left(\sqrt{2} , 0\right)$

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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 ${180}^{\circ}$ on the $x -$axis.

Apr 11, 2018

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

$y = m x + b$

where $m \ne 0$ has exactly one $y$-intercept and one $x$-intercept. If $b = 0$, then both the $x$ and $y$-intercepts are at the origin $\left(0 , 0\right)$.