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Roots on a Graph

When a factor [tex]\( x-k \)[/tex] is raised to an odd power, the graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x=k \)[/tex].

When a factor [tex]\( x-k \)[/tex] is raised to an even power, the graph only touches the [tex]\( x \)[/tex]-axis at [tex]\( x=k \)[/tex].

Describe the graph of the function [tex]\( g(x) = (x-1)(x+4)^3(x+5)^2 \)[/tex].

The graph:
- [tex]\(\square\)[/tex] the axis at [tex]\((1,0)\)[/tex].
- [tex]\(\square\)[/tex] the axis at [tex]\((-4,0)\)[/tex].
- [tex]\(\square\)[/tex] the axis at [tex]\((-5,0)\)[/tex].

Sagot :

GinnyAnswer
Let's discuss the graph of the function [tex]\( g(x) = (x-1) (x+4)^3 (x+5)^2 \)[/tex] and determine at which points it crosses or touches the x-axis.

### Step-by-Step Solution:

1. First Term: [tex]\( (x-1) \)[/tex]
- The factor [tex]\( (x-1) \)[/tex] can be rewritten as [tex]\( x-1 \)[/tex].
- This factor is raised to the power of 1, which is an odd number.
- When a factor [tex]\( x-1 \)[/tex] is raised to an odd power, the graph crosses the x-axis at [tex]\( x = 1 \)[/tex].

2. Second Term: [tex]\( (x+4)^3 \)[/tex]
- The factor [tex]\( (x+4)^3 \)[/tex] can be rewritten as [tex]\( (x + 4) \)[/tex].
- This factor is raised to the power of 3, which is an odd number.
- When a factor [tex]\( x+4 \)[/tex] is raised to an odd power, the graph crosses the x-axis at [tex]\( x = -4 \)[/tex].

3. Third Term: [tex]\( (x+5)^2 \)[/tex]
- The factor [tex]\( (x+5)^2 \)[/tex] can be rewritten as [tex]\( (x + 5) \)[/tex].
- This factor is raised to the power of 2, which is an even number.
- When a factor [tex]\( x+5 \)[/tex] is raised to an even power, the graph only touches the x-axis at [tex]\( x = -5 \)[/tex].

Based on this analysis, we can conclude the following about the graph of [tex]\( g(x) \)[/tex]:

- The graph crosses the x-axis at [tex]\( (1, 0) \)[/tex].
- The graph crosses the x-axis at [tex]\( (-4, 0) \)[/tex].
- The graph touches the x-axis at [tex]\( (-5, 0) \)[/tex].

Thus, the completed statements describing the graph are:

- The graph crosses the axis at [tex]\( (1, 0) \)[/tex].
- The graph crosses the axis at [tex]\( (-4, 0) \)[/tex].
- The graph touches the axis at [tex]\( (-5, 0) \)[/tex].
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