To analyze the behavior of the graph of the function
\[ f(x) = (2x - 7)^7 (x + 3)^4, \]
we need to identify the x-intercepts and how the function behaves at these points. The x-intercepts occur where \(f(x) = 0\). This occurs when either factor is equal to zero.
Step 1: Identify the x-intercepts
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From the first factor: \[ 2x - 7 = 0 \implies 2x = 7 \implies x = \frac{7}{2}. \] The first x-intercept is at \(x = 3.5\).
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From the second factor: \[ x + 3 = 0 \implies x = -3. \] The second x-intercept is at \(x = -3\).
Thus, the x-intercepts of the function are at \(x = \frac{7}{2}\) and \(x = -3\).
Step 2: Analyze the behavior at the x-intercepts
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At \(x = -3\):
- The factor \((x + 3)^4\) contributes a multiplicity of 4 (which is even).
- Since the multiplicity is even, the graph touches the x-axis at this intercept and turns around. Therefore, the function will approach the x-axis from above on one side (as \(x\) approaches -3 from the left) and come back from below (as \(x\) approaches -3 from the right), resulting in the graph touching the x-axis and not crossing it.
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At \(x = \frac{7}{2}\):
- The factor \((2x - 7)^7\) contributes a multiplicity of 7 (which is odd).
- Since the multiplicity is odd, the graph will cross the x-axis at this intercept. This means that as \(x\) approaches \(\frac{7}{2}\) from the left, \(f(x)\) will be negative, and as \(x\) moves past \(\frac{7}{2}\) to the right, \(f(x)\) will be positive.
Summary of the Behavior
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At \(x = -3\): The graph touches the x-axis and does not cross it because the multiplicity is even (4). It follows the behavior of approaching the axis, touching it, and then reversing direction.
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At \(x = \frac{7}{2}\): The graph crosses the x-axis because the multiplicity is odd (7). It goes from negative values to positive values as it crosses the x-axis.
This analysis provides insight into how the graph behaves at each x-intercept, indicating the nature of the intercepts through their multiplicities.