Sure! Let's solve the problem step-by-step.
First, we consider the function:
[tex]\[ f(x) = (x-2)(x+1)(x-6) \][/tex]
### Finding the y-intercept
The y-intercept of a function is the point where the graph of the function intersects the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
To find the y-intercept, substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = (0 - 2)(0 + 1)(0 - 6) \][/tex]
[tex]\[ f(0) = (-2)(1)(-6) \][/tex]
[tex]\[ f(0) = 12 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is:
[tex]\[ 12 \][/tex]
### Finding the x-intercepts
The x-intercepts of a function are the points where the graph of the function intersects the x-axis. This occurs where the function is equal to zero:
[tex]\[ f(x) = 0 \][/tex]
Set each factor of the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ (x-2) = 0 \][/tex]
[tex]\[ x = 2 \][/tex]
[tex]\[ (x+1) = 0 \][/tex]
[tex]\[ x = -1 \][/tex]
[tex]\[ (x-6) = 0 \][/tex]
[tex]\[ x = 6 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ x = 2, -1, 6 \][/tex]
### Summary
Given the function [tex]\( f(x) = (x-2)(x+1)(x-6) \)[/tex],
- The [tex]\( y \)[/tex]-intercept is [tex]\( 12 \)[/tex].
- The [tex]\( x \)[/tex]-intercepts are [tex]\( 2, -1, 6 \)[/tex].
Therefore, the completed answer is:
- The [tex]\( y \)[/tex]-intercept is [tex]\( 12 \)[/tex].
- The [tex]\( x \)[/tex]-intercepts are [tex]\( 2, -1, 6 \)[/tex].
