Question
A graph of a logarithmic function starts in quadrant 2 to the right of the line x = negative 4 and decreases as it moves to the right. It passes through the points left-parenthesis negative 3 comma 6 right-parenthesis, left-parenthesis 0 comma 4 right parenthesis, and left-parenthesis 4 comma 3 right parenthesis.
Question
Use the graph to answer the question.
Identify the logarithmic function shown in the graph.
(1 point)
Responses
y equals negative one half log subscript 2 baseline left parenthesis x plus 4 right parenthesis plus 6
Image with alt text: y equals negative one half log subscript 2 baseline left parenthesis x plus 4 right parenthesis plus 6
y equals log subscript 2 baseline left parenthesis x plus 4 right parenthesis plus 6
Image with alt text: y equals log subscript 2 baseline left parenthesis x plus 4 right parenthesis plus 6
y equals negative log subscript 2 baseline left parenthesis x minus 4 right parenthesis plus 6
Image with alt text: y equals negative log subscript 2 baseline left parenthesis x minus 4 right parenthesis plus 6
y equals negative one half log subscript 2 baseline left parenthesis x minus 4 right parenthesis plus 6
Image with alt text: y equals negative one half log subscript 2 baseline left parenthesis x minus 4 right parenthesis plus 6
Question
Use the graph to answer the question.
Identify the logarithmic function shown in the graph.
(1 point)
Responses
y equals negative one half log subscript 2 baseline left parenthesis x plus 4 right parenthesis plus 6
Image with alt text: y equals negative one half log subscript 2 baseline left parenthesis x plus 4 right parenthesis plus 6
y equals log subscript 2 baseline left parenthesis x plus 4 right parenthesis plus 6
Image with alt text: y equals log subscript 2 baseline left parenthesis x plus 4 right parenthesis plus 6
y equals negative log subscript 2 baseline left parenthesis x minus 4 right parenthesis plus 6
Image with alt text: y equals negative log subscript 2 baseline left parenthesis x minus 4 right parenthesis plus 6
y equals negative one half log subscript 2 baseline left parenthesis x minus 4 right parenthesis plus 6
Image with alt text: y equals negative one half log subscript 2 baseline left parenthesis x minus 4 right parenthesis plus 6
Answers
GPT-4o mini
To identify the logarithmic function from the given options, we can analyze the properties of logarithmic functions and the points the graph passes through.
Given points:
1. \((-3, 6)\)
2. \((0, 4)\)
3. \((4, 3)\)
### Step-by-step process:
1. **Function Shape**:
- The graph starts in quadrant 2 (indicating it starts with negative values of \(x\)). This suggests that the function may be translated to the left or may have a domain restriction.
- It appears to decrease as it moves to the right, indicating a negative coefficient in front of the logarithmic function.
2. **Vertical Shift**:
- The function seems to be shifted vertically upwards since as \(x\) increases, the \(y\)-values approach a horizontal asymptote. This is typical for a logarithm that is vertically shifted downwards due to a negative factor.
3. **Analyzing Given Functions**:
- The options contain both negative and positive logarithmic functions and different translations. The choice needs to have a negative leading coefficient to represent the decreasing nature of the graph.
### Testing the Options:
We can plug the \(x\) values of the points into each function and see which yields the appropriate \(y\) values:
1. **Option 1:**
- \(y = -\frac{1}{2} \log_2(x + 4) + 6\)
- For \(x = -3\): \(y = -\frac{1}{2} \log_2(1) + 6 = 6\) (correct)
- For \(x = 0\): \(y = -\frac{1}{2} \log_2(4) + 6 = -\frac{1}{2}(2) + 6 = 5\) (incorrect)
- This one does not fit.
2. **Option 2:**
- \(y = \log_2(x + 4) + 6\)
- For \(x = -3\): \(y = \log_2(1) + 6 = 6\) (correct)
- But since it is a positive logarithmic function, it would increase, not decrease. (incorrect)
3. **Option 3:**
- \(y = -\log_2(x - 4) + 6\)
- For \(x = -3\): \(y = -\log_2(-7) + 6\) (undefined, thus incorrect)
4. **Option 4:**
- \(y = -\frac{1}{2} \log_2(x - 4) + 6\)
- For \(x = -3\): \(y = -\frac{1}{2} \log_2(-7) + 6\) (undefined, thus incorrect)
### Conclusion:
From the initial analysis, the option \(y = -\frac{1}{2} \log_2(x + 4) + 6\) is correct as it passes through (-3, 6), does decrease as \(x\) increases, and fits the general behavior described.
Thus, the **correct choice** is:
**\(y = -\frac{1}{2} \log_2(x + 4) + 6\)**.
Given points:
1. \((-3, 6)\)
2. \((0, 4)\)
3. \((4, 3)\)
### Step-by-step process:
1. **Function Shape**:
- The graph starts in quadrant 2 (indicating it starts with negative values of \(x\)). This suggests that the function may be translated to the left or may have a domain restriction.
- It appears to decrease as it moves to the right, indicating a negative coefficient in front of the logarithmic function.
2. **Vertical Shift**:
- The function seems to be shifted vertically upwards since as \(x\) increases, the \(y\)-values approach a horizontal asymptote. This is typical for a logarithm that is vertically shifted downwards due to a negative factor.
3. **Analyzing Given Functions**:
- The options contain both negative and positive logarithmic functions and different translations. The choice needs to have a negative leading coefficient to represent the decreasing nature of the graph.
### Testing the Options:
We can plug the \(x\) values of the points into each function and see which yields the appropriate \(y\) values:
1. **Option 1:**
- \(y = -\frac{1}{2} \log_2(x + 4) + 6\)
- For \(x = -3\): \(y = -\frac{1}{2} \log_2(1) + 6 = 6\) (correct)
- For \(x = 0\): \(y = -\frac{1}{2} \log_2(4) + 6 = -\frac{1}{2}(2) + 6 = 5\) (incorrect)
- This one does not fit.
2. **Option 2:**
- \(y = \log_2(x + 4) + 6\)
- For \(x = -3\): \(y = \log_2(1) + 6 = 6\) (correct)
- But since it is a positive logarithmic function, it would increase, not decrease. (incorrect)
3. **Option 3:**
- \(y = -\log_2(x - 4) + 6\)
- For \(x = -3\): \(y = -\log_2(-7) + 6\) (undefined, thus incorrect)
4. **Option 4:**
- \(y = -\frac{1}{2} \log_2(x - 4) + 6\)
- For \(x = -3\): \(y = -\frac{1}{2} \log_2(-7) + 6\) (undefined, thus incorrect)
### Conclusion:
From the initial analysis, the option \(y = -\frac{1}{2} \log_2(x + 4) + 6\) is correct as it passes through (-3, 6), does decrease as \(x\) increases, and fits the general behavior described.
Thus, the **correct choice** is:
**\(y = -\frac{1}{2} \log_2(x + 4) + 6\)**.