c807429608
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.
2.7 KiB
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X and Y Intercepts
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(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) 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.
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).
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 you cannot 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, 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 it is easy to come up with graphs that even have infinitely many x-intercepts and infinitely many y-intercepts, such as r = theta.