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

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

Describe the graph of the function \(g(x) = (x-1)(x+4)^3(x+5)^2\):

The graph:

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



Answer :

GinnyAnswer
Let's analyze the function \( g(x) = (x - 1)(x + 4)^3(x + 5)^2 \) by examining each factor.

1. Factor \((x - 1)\):
- The exponent of this factor is \(1\), which is odd.
- When a factor is raised to an odd power, the graph crosses the x-axis at the corresponding root.
- Thus, the graph crosses the x-axis at \( x = 1 \).
- Point of intersection: \( (1, 0) \).

2. Factor \((x + 4)^3\):
- The exponent of this factor is \(3\), which is odd.
- When a factor is raised to an odd power, the graph crosses the x-axis at the corresponding root.
- Thus, the graph crosses the x-axis at \( x = -4 \).
- Point of intersection: \( (-4, 0) \).

3. Factor \((x + 5)^2\):
- The exponent of this factor is \(2\), which is even.
- When a factor is raised to an even power, the graph only touches the x-axis at the corresponding root without crossing it.
- Thus, the graph touches the x-axis at \( x = -5 \).
- Point of intersection: \( (-5, 0) \).

Based on this analysis, we can describe the graph of the function \( g(x) \):

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

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