To identify the logarithmic function based on the given information about the function's behavior and points it passes through, we can analyze each possible response.
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Behavior of the Function: The graph starts in quadrant 2 (so it exists for negative x-values before going to the right) and decreases as it moves to the right. This indicates that there should be a negative coefficient for the logarithm in the function.
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Vertical Shift: The function passes through the point \((-3, 6)\), which means that when \(x = -3\), \(y = 6\).
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Transformation: The \(x\)-values in the logarithmic arguments suggest a leftward shift, as \((x + 4)\) indicates that the vertical asymptote (the value of \(x\) where the logarithmic function is undefined) is at \(x = -4\).
Now let's evaluate the given options based on these characteristics.
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Option 1: \(y = -\frac{1}{2} \log_2(x + 4) + 6\)
- This option has a negative coefficient and it is shifted up by 6. It can potentially pass through the points but we need to verify.
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Option 2: \(y = \log_2(x + 4) + 6\)
- This function has a positive coefficient, suggesting it will increase rather than decrease. So this option is not correct.
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Option 3: \(y = -\log_2(x - 4) + 6\)
- The argument \(x - 4\) implies a vertical asymptote at \(x = 4\), which does not match the description; thus, this option is not correct.
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Option 4: \(y = -\frac{1}{2} \log_2(x - 4) + 6\)
- Similar to Option 3, this has the wrong argument and asymptote position, making it incorrect as well.
Now let's test Option 1 with the point \((-3, 6)\):
- Plugging in \(-3\) into the function: \[ y = -\frac{1}{2} \log_2(-3 + 4) + 6 = -\frac{1}{2} \log_2(1) + 6 = -\frac{1}{2}(0) + 6 = 6 \]
- The point is satisfied.
Now let's check another point \((0, 4)\):
- Plugging in \(0\): \[ y = -\frac{1}{2} \log_2(0 + 4) + 6 = -\frac{1}{2} \log_2(4) + 6 = -\frac{1}{2}(2) + 6 = -1 + 6 = 5 \]
- This does not satisfy the point \( (0, 4) \).
Lastly, let's try checking for the point \((4, 3)\):
- Plugging in \(4\): \[ y = -\frac{1}{2} \log_2(4 + 4) + 6 = -\frac{1}{2} \log_2(8) + 6 = -\frac{1}{2}(3) + 6 = -\frac{3}{2} + 6 = 4.5 \]
- This doesn’t return correct values either.
Given these calculations, none of the functions properly pass through all three key points. They also bear wrong transformations and asymptotes that contradict the start and behavior.
Please double-check the graphical accuracy on the pip chart, or consider that the problem may have errors in expected correct transformations or placements if the graphs contradicted real world behavior projections through the stated coordinates.
However, based on logical deductions made: The potential candidate still appears to be Option 1: \(y = -\frac{1}{2} \log_2(x + 4) + 6\) as function alterations need grading for punctual representation errors across original graphical data standards.
Nonetheless, none convincingly provide correctness - I suggest reviewing the underlying graph points again or considering progression differently.