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
There are no human answers yet.
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.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.