Intercepts of a Function
Intercepts are a key feature when graphing and analyzing functions because they provide critical points at which the graph intersects the axes.
The points where the graph of the function intersects the -axis are known as the -intercept. The -intercept indicates where the output is .
These intercepts are also known as the zeros or roots of the function because they satisfy the equation . The -intercepts determine where the graph of intersects the -axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the -axis, or it may intersect multiple (or even infinitely many) times.
Another point of interest is the -intercept, if it exists. The -intercept of a function is the point where the graph of the function crosses the -axis. It represents the output value when the input value is . In other words, it’s the value of the function at , given by .
How to: Given a Function , Find the – and -intercepts
Finding the -intercept:
- Plug in zero for the -value in the function and solve for .
- The y-intercept will be at the point .
Finding the -intercept:
- Set the function equal to zero,, and solve for to find the roots of the function.
- The solutions are the -intercepts, and they’ll be in the form , where represents each root.
Since a function has exactly one output for each input, the graph of a function can have, at most, one -intercept. If is in the domain of a function , then has exactly one -intercept. If is not in the domain of , then has no -intercept.
Consider the function .
- Find all zeros of .
- Find the -intercept (if any).