Answered

Join the growing community of curious minds on IDNLearn.com and get the answers you need. Ask any question and get a detailed, reliable answer from our community of experts.

Describe the graph of the function at its roots.

[tex]\[ f(x)=(x-2)^3(x+6)^2(x+12) \][/tex]

At [tex]\( x=2 \)[/tex], the graph [tex]\(\square\)[/tex] the [tex]\( x \)[/tex]-axis.

At [tex]\( x=-6 \)[/tex], the graph [tex]\(\square\)[/tex] the [tex]\( x \)[/tex]-axis.

At [tex]\( x=-12 \)[/tex], the graph [tex]\(\square\)[/tex] the [tex]\( x \)[/tex]-axis.

Sagot :

GinnyAnsweridn
To describe the behavior of the graph of the function [tex]\( f(x) = (x-2)^3(x+6)^2(x+12) \)[/tex] at its roots, let's analyze each root in turn.

1. At [tex]\( x = 2 \)[/tex]:
- The term [tex]\((x-2)^3\)[/tex] contributes to the function becoming zero at [tex]\( x = 2 \)[/tex].
- The exponent 3 is an odd number, which means that at this root, the graph will touch the x-axis and change direction (cross the x-axis).

Therefore, at [tex]\( x=2 \)[/tex], the graph touches and crosses the x-axis.

2. At [tex]\( x = -6 \)[/tex]:
- The term [tex]\((x+6)^2\)[/tex] results in the function becoming zero at [tex]\( x = -6 \)[/tex].
- The exponent 2 is an even number, implying that at this root, the graph will touch the x-axis but will not change direction (it does not cross the x-axis).

Hence, at [tex]\( x=-6 \)[/tex], the graph touches but does not cross the x-axis.

3. At [tex]\( x = -12 \)[/tex]:
- The term [tex]\((x+12)\)[/tex] causes the function to be zero at [tex]\( x = -12 \)[/tex].
- The exponent of this term is 1, which is an odd number. This means that at this root, the graph will touch the x-axis and change direction (cross the x-axis).

Consequently, at [tex]\( x=-12 \)[/tex], the graph touches and crosses the x-axis.

Summarizing our findings:
- At [tex]\( x=2 \)[/tex], the graph touches and crosses the x-axis.
- At [tex]\( x=-6 \)[/tex], the graph touches but does not cross the x-axis.
- At [tex]\( x=-12 \)[/tex], the graph touches and crosses the x-axis.

So the completed statement is:

At [tex]\( x=2 \)[/tex], the graph touches and crosses the [tex]\( x \)[/tex]-axis.

At [tex]\( x=-6 \)[/tex], the graph touches but does not cross the [tex]\( x \)[/tex]-axis.

At [tex]\( x=-12 \)[/tex], the graph touches and crosses the [tex]\( x \)[/tex]-axis.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.