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 [latex]x[/latex]-axis are known as the [latex]x[/latex]-intercept. The [latex]x[/latex]-intercept indicates where the output [latex]f(x)[/latex] is [latex]0[/latex].
These intercepts are also known as the zeros or roots of the function because they satisfy the equation [latex]f(x)=0[/latex]. The [latex]x[/latex]-intercepts determine where the graph of [latex]f[/latex] intersects the [latex]x[/latex]-axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the [latex]x[/latex]-axis, or it may intersect multiple (or even infinitely many) times.
Another point of interest is the [latex]y[/latex]-intercept, if it exists. The [latex]y[/latex]-intercept of a function is the point where the graph of the function crosses the [latex]y[/latex]-axis. It represents the output value when the input value [latex]x[/latex] is [latex]0[/latex]. In other words, it’s the value of the function [latex]f(x)[/latex] at [latex]x=0[/latex], given by [latex](0,f(0))[/latex].
How to: Given a Function [latex]f\left(x\right)[/latex], Find the [latex]y[/latex]– and [latex]x[/latex]-intercepts
Finding the [latex]y[/latex]-intercept:
- Plug in zero for the [latex]x[/latex]-value in the function and solve for [latex]f(0)[/latex].
- The y-intercept will be at the point [latex](0,f(0))[/latex].
Finding the [latex]x[/latex]-intercept:
- Set the function equal to zero,[latex]f(x)=0[/latex], and solve for [latex]x[/latex] to find the roots of the function.
- The solutions are the [latex]x[/latex]-intercepts, and they’ll be in the form [latex](x,0)[/latex], where [latex]x[/latex] represents each root.
Since a function has exactly one output for each input, the graph of a function can have, at most, one [latex]y[/latex]-intercept. If [latex]x=0[/latex] is in the domain of a function [latex]f[/latex], then [latex]f[/latex] has exactly one [latex]y[/latex]-intercept. If [latex]x=0[/latex] is not in the domain of [latex]f[/latex], then [latex]f[/latex] has no [latex]y[/latex]-intercept.
Consider the function [latex]f(x)=-4x+2[/latex].
- Find all zeros of [latex]f[/latex].
- Find the [latex]y[/latex]-intercept (if any).