From c807429608930d5f623b29758371debb8213132d Mon Sep 17 00:00:00 2001 From: Alexander Molnar <37451552+BTmathic@users.noreply.github.com> Date: Wed, 17 Apr 2019 07:41:01 -0400 Subject: [PATCH] Fixed errors and added examples with variety (#24300) Made precise that intercepts are (x,y)-points and not numbers. Added more examples to show different possibilities that can occur and removed the oversized image of a few short lines of equations. --- .../mathematics/x-and-y-intercepts/index.md | 15 +++++++++------ 1 file changed, 9 insertions(+), 6 deletions(-) diff --git a/guide/english/mathematics/x-and-y-intercepts/index.md b/guide/english/mathematics/x-and-y-intercepts/index.md index 6e839bab51..5f81fad7b1 100644 --- a/guide/english/mathematics/x-and-y-intercepts/index.md +++ b/guide/english/mathematics/x-and-y-intercepts/index.md @@ -3,15 +3,18 @@ title: X and Y Intercepts --- ## X and Y Intercepts -The X intercept(s) are the point(s) on the X axis where the graph of the function crosses or touches the axis. They occur where ```y = 0```. +Consider a function `y = f(x)` and its graph. The `x`-intercept(s) of `f(x)` are the point(s) where the graph of the function crosses or touches the `x`-axis, i.e., they are points of the form `(x,0)` occuring when `f(x) = y = 0`. -The Y intercept is the point at which the graph of the function crosses the Y axis. It occurs where ```x = 0```. +The `y`-intercept(s) are the point(s) at which the graph of the function crosses the `y` axis, i.e., they are points of the form `(0,y)` occuring when `y = f(0)`. A function can have multiple X intercepts, but can only have a single Y intercept. Why is only a single Y intercept allowed? By definition, a function can only have one output, y, for each unique input, x. Thus, a graph would violate this definition if ```x = 0``` produced more than one output, y. -To find the X intercept(s), set ```y = 0``` and solve the equation for x. +To find the `x`-intercept(s) of a function, set `y = 0` and solve the equation for `x`. To find the `y` intercept, set `x = 0` and solve the equation for `y`. -To find the Y intercept, set ```x = 0``` and solve the equation for y. +For the simplest case, consider a line `y = mx + b`. To find the `x`-intercepts we set `y=0`, giving `mx + b = 0`, which can be solved as `x = -b/m`, so the (only) `x`-intercept is the point `(-b/m,0)`. To find the `y`-intercept we set `x=0` and have `y = 0 + b = b`, hence the `y`-intercept is `(0,b)`. -As an example, the X and Y intercepts have been solved for a typical line in the form ```y = mx + b```. -![An image demonstrating finding the x and y intercepts for a typical line in the form of y = mx + b . The Y intercept is equal to b and the X intercept is equal to negative b divided by m](https://raw.githubusercontent.com/hayleycd/images/master/solve_intercepts.jpeg) +With more complex functions it may not be possible to solve for the `x`-intercepts algebraically, e.g., with the function [`y = x^5 - x - 1`](https://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E5+-+x+-+1) you [cannot](https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem) solve for `x`. However, looking at the graph can tell you how many `x`-intercepts there are and with a calculator/computer you can get an approximation of each one. On the other hand, the `y`-intercept is always easy to find (if it exists), just set `x=0` and solve (if possible). + +Note that even if a graph is not given by a function, e.g., the circle of radius 1, given by [`x^2 + y^2 = 1`](https://www.wolframalpha.com/input/?i=plot+x%5E2+%2B+y%5E2+%3D+1), you can still find the `x` and `y`-intercepts by plugging in `y=0` and `x=0`, and solving for `x` and `y`, respectively. For the circle we see the `x`-intercepts occur when `x^2 + 0 = 1`, i.e., `(-1,0)` and `(1,0)`. Similarly, the `y`-intercepts are `(0,-1)` and `(0,1)`. + +If we do not limit ourselves to graphs of functions of `x`, then a graph can have any numbers of `y`-intercepts, e.g., a polynomial `x = f(y)` of degree `n` has (at most) `n` `y`-intercepts, and the graph of `x = sin(y)` has infinitely many `y`-intercepts. Using [polar coordinates](https://en.wikipedia.org/wiki/Polar_coordinate_system) it is easy to come up with graphs that even have infinitely many `x`-intercepts and infinitely many `y`-intercepts, such as [`r = theta`](https://www.wolframalpha.com/input/?i=plot+r+%3D+theta).