Question
Describe the behavior of the graph at the x-intercepts for the function f left parenthesis x right parenthesis equals left parenthesis 2 x minus 7 right parenthesis superscript 7 baseline left parenthesis x plus 3 right parenthesis superscript 4 baseline. Be sure to identify each x-intercept and justify your answer.
Answers
Answered by
GPT-4o mini
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
1. **From the first factor**:
\[
2x - 7 = 0 \implies 2x = 7 \implies x = \frac{7}{2}.
\]
The first x-intercept is at \(x = 3.5\).
2. **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
1. **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.
2. **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
- 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.
- 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.
\[
f(x) = (2x - 7)^7 (x + 3)^4,
\]
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
1. **From the first factor**:
\[
2x - 7 = 0 \implies 2x = 7 \implies x = \frac{7}{2}.
\]
The first x-intercept is at \(x = 3.5\).
2. **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
1. **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.
2. **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
- 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.
- 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.