To analyze the behavior of the graph at the x-intercepts for the function \( f(x) = (2x - 7)^7 (x + 3)^4 \), we first need to identify the x-intercepts by setting \( f(x) = 0 \) and solving for \( x \).
Finding the X-Intercepts
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The x-intercepts occur when \( f(x) = 0 \).
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The function can be zero when either factor is zero:
- \( 2x - 7 = 0 \) leads to \( x = \frac{7}{2} = 3.5 \)
- \( x + 3 = 0 \) leads to \( x = -3 \)
Thus, the x-intercepts are at \( x = 3.5 \) and \( x = -3 \).
Behavior at the X-Intercepts
Next, we analyze the behavior of the function at each x-intercept by considering the multiplicity of each intercept:
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At \( x = 3.5 \):
- The factor \( (2x - 7) \) has a multiplicity of 7 (because it is raised to the power of 7).
- An odd multiplicity means that as \( x \) approaches \( 3.5 \), the function will cross the x-axis.
- Therefore, the behavior of \( f(x) \) near \( x = 3.5 \):
- As \( x \) approaches \( 3.5 \) from the left (\( x < 3.5 \)), \( f(x) \) will be negative.
- As \( x \) approaches \( 3.5 \) from the right (\( x > 3.5 \)), \( f(x) \) will be positive.
- Conclusion: The graph crosses the x-axis at \( x = 3.5 \).
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At \( x = -3 \):
- The factor \( (x + 3) \) has a multiplicity of 4 (because it is raised to the power of 4).
- An even multiplicity means that as \( x \) approaches \( -3 \), the function will touch the x-axis but not cross it.
- Therefore, the behavior of \( f(x) \) near \( x = -3 \):
- As \( x \) approaches \( -3 \) from the left (\( x < -3 \)), \( f(x) \) will be positive.
- As \( x \) approaches \( -3 \) from the right (\( x > -3 \)), \( f(x) \) will also be positive.
- Conclusion: The graph touches the x-axis at \( x = -3 \) and does not cross.
Summary of Behavior
- At \( x = 3.5 \): The graph crosses the x-axis due to the odd multiplicity of 7.
- At \( x = -3 \): The graph touches the x-axis and does not cross due to the even multiplicity of 4.
This analysis helps us understand how the function behaves around its x-intercepts.