To analyze the transformations from \( b(x) \) to \( c(x) \), we start with the definitions:
- \( b(x) = 1.54 - \sqrt{x} - 1 \)
- \( c(x) = 9 - \sqrt{x} - 1 \)
Let's simplify both functions:
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For \( b(x) \): \[ b(x) = 1.54 - \sqrt{x} - 1 = 0.54 - \sqrt{x} \]
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For \( c(x) \): \[ c(x) = 9 - \sqrt{x} - 1 = 8 - \sqrt{x} \]
Now we can compare the two functions:
- The general form of both functions is \( y = A - \sqrt{x} \), where \( A \) is a constant.
- In \( b(x) \), the constant term is \( 0.54 \).
- In \( c(x) \), the constant term is \( 8 \).
Finding the Factors of Transformation
Next, we analyze the vertical change caused by the difference in the constant terms:
- To determine how much \( c(x) \) stretches vertically compared to \( b(x) \), we look at the difference in the constants:
\[ \Delta A = c(0) - b(0) = (8 - \sqrt{0}) - (0.54 - \sqrt{0}) = 8 - 0.54 = 7.46 \]
The transformation involves finding the relationship between \( b(x) \) and \( c(x) \): \[ c(x) = \text{(a vertical stretch) + b(x)} \]
To analyze the nature of this transformation, we calculate:
\[ \text{Vertical change} = \frac{\text{New constant term}}{\text{Old constant term}} = \frac{8}{0.54} \approx 14.8148 \]
This doesn't match exactly with any of the options, so the easiest alternative is to directly state:
Summary
The transformation from \( b(x) \) to \( c(x) \) is a vertical stretch. The change corresponds to a factor greater than 8.
The provided options for the transformation (vertical stretch/compression by a factor of 6, 9, or 7.5), cannot be justified from \( b(x) \) to \( c(x) \) based on the calculations performed.
Conclusion
I suggest verifying the options again, as none of them matches the transformation we have discussed.
If you need any further clarification on the functions or the transformations, please let me know!